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Cengage Maths Solutions Class 12 Limits and Derivatives - Calculus

Cengage Maths Solutions Class 12 Limits and Derivatives (https://doubtnut.com/cengage-solutions/chapter-limits-and-derivatives-1) PDfs provides accurate solutions to practice before your final exam. Students can study this comprehensive Cengage Maths PDf Solutions on Limits and Derivatives and its various sub-topics for effective exam preparation. The Cengage 12 PDF solutions help the student to swiftly practice and revise the important concepts just before the annual exam to score high marks in maths.<br>For more such video solutions/Pdf solutions, visit https://doubtnut.com and clear your doubt instantly.

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Cengage Maths Solutions Class 12 Limits and Derivatives - Calculus

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  1. CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER LIMITS AND DERIVATIVES || CALCULUS  Download Doubtnut Today Ques No. Question CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits ( ) 1 If y = 2-2 1 1-x , then find lim x→1+y  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits Evaluate lim x→0-x2- 3x + 2 2 x3- 2x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits →0sinx - 2 cosx - 1 Evaluate (lim)x 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits → 5π 4[sinx + cosx], where [.] denotes the greatest integer function. Evaluate (lim)x 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits [ , where [.] denotes the greatest integer function. ] n 1 2r ∑ r=1 Evaluate lim n→ ∞ 5  Watch Free Video Solution on Doubtnut

  2. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of LimitsCENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits { |x - 4| x - 4,x ≠ 40,x = 4atx = 4 Evaluate the left-and right-hand limits of the function f(x) = |x − 4| 6 f(x) = { ,x ≠ 40, x = 4atx x − 4 = 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits 1 + x2, { Evaluate the left-and right-hand limits of the function defined by f(x) = if 0 ≤ x < f(x) = {1 + x2, if 0 ≤ x < 1 and 2 7 − x, at x = 1. Also, show that lim x→1f(x) does not exist if x > 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits Let f(x) = {cos[x], x ≥ 0|x| + a, x < 0 The find the value of a, so that (lim)x where [x] denotes the greatest integer function less than or equal to x . →0f(x) exists, 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits →-2+x2- 1 2x + 4 Evaluate: (lim)x 9  Watch Free Video Solution on Doubtnut

  3. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of LimitsCENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits [x - 2] log(x - 2) , where [.] represents the greatest integer function. 10 Evaluate : (lim)x→2+  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits Evaluate: (lim)x→0sin[cosx] 1 + [cosx]([.] denotes the greatest integer function). 11  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits { x - |x| x If f(x) = , x ≠ 02, x = 0, showthat(lim)x →0 x − |x| f(x) = { 12 , x ≠ 02,x = 0, x showthat(lim)x→ f(x) does not exist. 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits 1 x - 1 Show that (lim)x→0e does¬eξst 13 1 x + 1 e  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits 14

  4. 3x + |x| 7x - 5|x| Evaluate (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits { →0f(x)x2,x > 0 if eξsts f(x) = x,x ≤ 01,x = 0, thenf ∈ d(lim)x f(x) = {x,x ≤ 01, x = 0, thenf 15 0f(x)x2,x > 0 if eξsts ∈ d(lim)x→  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Concept Of Limits Consider the following graph of the function y = f(x). Which of the following is/are correct? fig. (lim)x →1f(x) does not exist. (lim)x →2f(x) does not exist. (lim)x →3f(x) = 3 16  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits →af(x)g(x). If (lim)x →a[f(x) + g(x)] = 2and(lim)x →a[f(x) - g(x)] = 1, then find the value of (lim)x 17  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits x + 3,x < 1,x2- 2x - 2,1 ≤ x < 2x - 5, x ≥ 2 { Let f(x) = {x + 1,x > 02 - x, x ≤ 0 and g(x) = g(x) = {x + 3, x < 1, x2− 2x − 2, 1 18 ≤ x < 2x − 5, x ≥ 2 Find the LHL and RHL of g(f(x)) at x = 0 and, hence, fin lim x→0g(f(x)).  Watch Free Video Solution on Doubtnut

  5. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of LimitsCENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits Evaluate: lim x→ ∞x + 7sinx -2x + 13 using sandwich theorem. 19  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits ( ) for all x ≠ 0, then find the value of (lim)x ( ) x2 12 x3 9 If 3 - ≤ f(x) ≤ 3 + →0f(x) 20  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits If [.] denotes the greatest integer function, then find the value of (lim)x [x] + [2x] + + [nx] n2 →0 21  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits ( ) Suppose that f is a function such that 2x2≤ f(x) ≤ x x2+ 1 for all x that are near to 1 but not equal to 1. Show that this fact contains enough information for us to find (lim)x 22 →1f(x). Also, find this limit.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits 1 1 n Evaluate : (lim)x→ ∞ 1 + n2+ 2 + n2+ ... + 23 n + n2  Watch Free Video Solution on Doubtnut

  6. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of LimitsCENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits → R satisfies the inequality `(cos(2x-4)-33)/2 24 Let f:(1, 2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits x2+ x - 2 x + 3 x = - 1and(lim)x x2+ 2x - 1 x + 3 f(x) x2≤ →1f(x)eξsts then find the value of (lim)x If hold for a certain interval containing the point ≤ 25 →-1f(x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits lim x→ ∞[x] Evaluate the following limits using sandwich theorem: x,where[.] 26 represents greatest integer function.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Algebra Of Limits (log)ex x Evaluate the following limits using sandwich theorem: (lim)x 27 → ∞  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Use Of Expansions In Evaluating Limits 5sinx - 7sin2x + 3sin3x x2sinx (lim)x →0 28  Watch Free Video Solution on Doubtnut

  7. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Use Of ExpansionsCENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Use Of Expansions In Evaluating Limits 1 x - e + 1 2ex (1 + x) 29 Evaluate: (lim)x →0 x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Use Of Expansions In Evaluating Limits { } x3 sinx - x + 30 6 Evaluate the limits using the expansion formula of functions (lim)x →0 x5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Use Of Expansions In Evaluating Limits sinx + log(1 - x) x2 31 Evaluate the limits using the expansion formula of functions (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Use Of Expansions In Evaluating Limits ex- 1 - x x2 32 Evaluate the limits using the expansion formula of functions (lim)x →0  Watch Free Video Solution on Doubtnut

  8. CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of x2- 5x + 6 x2- 4 33 Evaluate : (lim)x →2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of ( ) 2 1 34 Evaluate : (lim)x 1 - x2+ →1 x - 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of x2+ x(log)ex - (log)ex - 1 ( Evaluate : (lim)x 35 →1 ) x2- 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of 1 - sin2x 1 + cos4x 36 →π 4 Evaluate (lim)x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of

  9. 1 - cot3x 2 - cotx - cot3x 37 → π 4 Evaluate : (lim)x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of √2 + x -√2 38 Evaluate (lim)x→0 x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits ( AND DERIVATIVES_Evaluation Of →0sinx + (log)e√1 + sin2x - sinx The value of (lim)x sin3x 0(sinx 39 (lim)x→ √1 + sin2x − sinx + (log)e sin3x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of x10- 1024 x5- 32 40 Evaluate (lim)x →2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of √√ √ √x +√√x +√√x+ √√x - 4 41 Evaluate: (lim)x →1 x - 1  Watch Free Video Solution on Doubtnut

  10. CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of 5 3 - (a + 2) x - a 5 Evaluate: lim x→a(x + 2) 3 42  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of xn- 2n x - 2 xn− 2n = 80amdm ∈ N, thenf ∈ dthevalueofn. If (lim)x →2 43 (lim)x→ = 80amdm ∈ N, x − 2 2 thenf ∈ dthevalueofn.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of √(x + 7) - 3√(2x - 3) 44 Evaluate: (lim)x →2 (x + 63 - 2(3x - 5)3)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of ax2+ bx + c dx2+ ex + f 45 . Evaluate : (lim)x → ∞  Watch Free Video Solution on Doubtnut

  11. CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of ( ) 2πn 3n + 1 46 → ∞sinn , n ∈ N. Evaluate: (lim)x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of ∞√3x2- 1 -√2x2- 1 4x + 3 47 Evaluate: (lim)x →  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of → ∞√x√x + c -√x ( ) 48 Evaluate : (lim)x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of Evaluate: lim x→ ∞√x2+ 1 - x3+ 13 49 x4+ 14 - x4+ 15  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of

  12. → ∞ √25x2- 3x + 5x ( ) 50 Evaluate : (lim)x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of ( ) 3x2+x x2+ x - 1 3x2+ 2x + 4 x-2 51 Evaluate: lim x→ ∞  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of x2+√1 + x4- x√2 {√ } → ∞x3 52 Evaluate: (lim)x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of 3√(x + 1)(x + 2)(x + 3) - x 53 Evaluate: lim x→ ∞  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of 1 → ∞4n+ 5n ( ) 54 Evaluate: (lim)n n  Watch Free Video Solution on Doubtnut

  13. CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of nψn2(n!) n + 1 55 Evaluate: (lim)n → ∞  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of → ∞( - 1)n-1sin π√n2+ 0.5n + 1 , wheren ∈ N ( ) Evaluate: (lim)n ∞( − 1)n− 1sin(π√n2+ 0.5n + 1), 56 (lim)n− → wheren ∈ N  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of Let the sequence `<>realνmberssatiiestherecurrencerelation realνmberssatiiestherecurrencerelation b_(n+1)=1/3(2b_n+(125)/(b n2)),b_n!=0.Thenf ∈ dthe("lim")_(nvecoo)b_ndot` 57  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of 1 n3 12x + {[ ] If [x] denotes the greatest integer less than or equal to x, then evaluate (lim)n → ∞ 1 58 {[12x] + [22x] + [32x] (lim)n− → ∞ n3 + }[n2x]}  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of ( ) (2x - 3)√x - 1 2x2 59 Evaluate the limit: lim x→1  Watch Free Video Solution on Doubtnut

  14. CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of ∑k = 1100xk- 100 x - 1 60 Evaluate the limit: (lim)x →1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of → ∞ √a2x2+ ax + 1 -√a2x2+ 1 [ ] Evaluate the limit: (lim)x ∞[√a2x2+ ax + 1 61 (lim)x− → −√a2x2+ 1]  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of √3x - a -√x + a 62 Evaluate the limit: (lim)x→a x - a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of ( 12- 22+ 33- 42+ 52+ nterms n2 Evaluate the limit: (lim)n→ ∞

  15. 63 (lim)n→ ∞ 12− 22+ 33− 42+ 52+ nterms ( n2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of [ ] 1 1 64 Evaluate the limit: (lim)h h8 + h3- →0 2h  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of √2 -√1 + cosx sin2x Evaluate the limit: (lim)x 65 →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of ( ) Evaluate the limit: (lim)n→ ∞cos π√n2+ n when n is an integer 66  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Algebraic Limits AND DERIVATIVES_Evaluation Of

  16. 4 + 3an 3 + 2an 4 + 3an If a1= 1andan+ 1= ,n ≥ 1,and if (lim)n → ∞an= a, 67 a1= 1andan+1= , n ≥ 1,and 3 + 2an if (lim)n− → then find the value of a. ∞an= a,  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of sin3x x 68 Evaluate the limit (lim)x→0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of sinax sinbx 69 Evaluate the limit: (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of sin(logx) logx 70 Evaluate the limit: (lim)x →1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of ( ) 1 xsin-1 2x 71 Evaluate: (lim)x →0 1 + x2  Watch Free Video Solution on Doubtnut

  17. CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of 1 - cos2x x2 72 Evaluate: (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of tanx - sinx x3 73 Evaluate: (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of 1 + cos2x (π - 2x)2 π 2 Evaluate: lim x→ 74  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of ( ) a 2x 75 → ∞2x-1tan Evaluate : (lim)x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of 76

  18. x2- x - 2 Evaluate: lim x→2 x2- 2x - sin(x - 2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of ( ( ) ) Evaluate: lim x→ ∞x tan-1x + 1 π 4 77 - x + 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of sinθ θ - 1 prove that the area of circle of radius R is πR2 (Figure) 78 Using (lim)θ →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of sinx x Evaluate: (lim)x ,where[.] represents the greatest integer function. 79 →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Trigonometric Limits AND DERIVATIVES_Evaluation Of [ , where [.]represent the greatest integer function ] tan-1x x 80 Evaluate: (lim)x→0

  19.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of sinx0 x 81 Evaluate: (lim)x→0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of 1 - cosmx 1 - cosnx 82 Evaluate: (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of √2cosx - 1 83 →π 4 Evaluate: (lim)x cotx - 1  Watch Free Video Solution on Doubtnut

  20. CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of cot2x - cosec2x x Evaluate: (lim)x 84 →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of Evaluate: lim x→0tan2x - x 85 3x - sinx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of 2√3sin [ ( π 6+ h - cos ) ( ( )] π 6+ h Evaluate: (lim)h →0 ) √3h√3cosh - sinh 86 (lim π π 2[√3sin( + h) − cos( + h)] 6 6 )h→ 0 √3h(√3cosh − sinh)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of Evaluate: lim n→ ∞ncos ( ( ) ( ) π 4n π 4n 87 sin  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of → 0, along the curve x = y2 y2+ sinx x2+ siny2 where (x, y) 88 Evaluate: (lim)x →0x →0  Watch Free Video Solution on Doubtnut

  21. CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of ( ) 1-x2 1+ x2 cos-1 89 Evaluate: (lim)x →0 sin- 1x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of tanx x 90 Evaluate: (lim)x where [.] represents the greatest integer function →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of tanx x 91 Evaluate: (lim)x where [.] represents the greatest integer function →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of 2x- 1 √1 + x - 1 Evaluate: (lim)x 92 →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Evaluation Of

  22. Exponential And Logarithmic Limits ax-1- 1 sinπx 93 Evaluate: (lim)x →1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of 10x- 2x- 5x+ 1 xtanx Evaluate: (lim)x 94 →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of 32x- 23x x 95 Evaluate: (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of x - 2 96 Evaluate: (lim)x →2 (log)a(x - 1)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of logx - loga x - a 97 Evaluate: (lim)x →a  Watch Free Video Solution on Doubtnut

  23. CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of log(5 + x) - log(5 - x) x 98 Evaluate: (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of Leg pn= aPn-1- 1, ∀n = 2, 3,, andletP1= ax- 1, pn= aPn−1− 1, ∀n = 2,3,,andletP1 = ax− 1, 99 Pn x where a ∈ R+. Then evaluate (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of → ∞x a [ ( )] 1 x - 1 100 Evaluate: (lim)x ,a > 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of x2x- x 1 - cosx 101 Evaluate: (lim)x →0  Watch Free Video Solution on Doubtnut

  24. CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of sin ex-2- 1 ( ) 102 Evaluate: lim x→2 log(x - 1)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of ex^ 2 - cosx x2 Evaluate: (lim)x →0 103  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of ex+ e-x- 2 x2 104 Evaluate: (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of log(x - a) ( Evaluate: (lim)x →a 105 ) log ex- ea  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of atanx- asinx tanx - sinx, a > 0 106 Evaluate: (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of 1 - 3x- 4x+ 12x ( ) 107 Evaluate: (lim)x →0 √(2cosx + 7) - 3  Watch Free Video Solution on Doubtnut

  25. CENGAGE_MATHS_CALCULUS_LIMITS Exponential And Logarithmic Limits AND DERIVATIVES_Evaluation Of (729)x- (243)x- (811)x+ 9x+ 3x- 1 x3 Evaluate: (lim)x 108 →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim 1 109 Evaluate: (lim)x → ∞x x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim →π 2(cosx)cosx Evaluate: (lim)x 110  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim →0(1 + x)osecxc 111 Evaluate: (lim)x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim Evaluate: (lim)x→0(cosx)cotx 112

  26.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim )( ) sinx ( sinx x 113 x-sinx Evaluate (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim ( ) 2 ax+ bx+ cx 3 x; (a, b, c > 0) 114 Evaluate: (lim)x→0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim ( ))( ( )) ( ( ) ( x 2 x x xn]1/x If then find f(n) = (lim)x→0{ 1 + sin 1 + sin ...... 1 + sin 115 x2 → ∞f(n). (lim)n  Watch Free Video Solution on Doubtnut

  27. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The FormCENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim The population of a country increases by 2% every year. If it increases k times in a century, then prove that [k] = 7,where[.] represents the greatest integer function. 116  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim ( ) x + 2 x + 1 x+3 117 Evaluate the limit: (lim)x → ∞  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim ( ) 1 c+ dx 118 Evaluate: lim x→ ∞ 1 + , where a, b,c and dare positive a + bx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim ( ) Evaluate: (lim)x→ ∞1 +1 x 119 x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim

  28. 2x2- 9x + 8 ( ) 120 cot(2x-7) →7 2 Evaluate: (lim)x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Limits Of The Form Lim ( ) π { ( )} sec2 π 121 2-qx Evaluate: lim x→0 sin2 2 - px  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits Let f(x) be a twice-differentiable function and f(0) = 2. The evaluate: 2f(x) - 3f(2x) + f(4x) x2 122 (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits Let f(a) = g(a) = k and their nth derivatives exist and are not equal for some n. If. f(a)g(x) - f(a) - g(a)f(x) + g(a) g(x) - f(x) 123 (lim)x = 4 →a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits If the graph of the function y = f(x) has a unique tangent at the point (a, 0) through (log)e{1 + 6f(x)} 3f(x) 124 which the graph passes, then evaluate (lim)x →a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits ( ) Evaluate : lim x→0(log)tan2xtan22x. 125  Watch Free Video Solution on Doubtnut

  29. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule ForCENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits 126 Evaluate `("lim")_(xvecoo)x(log)_e{(sin(a+1/x))/(sina)},0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits Evaluate: lim x→0xm(logx)n, m, n ∈ N. 127  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits sin-1x - tan- 1x x3 128 Evaluate: (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits xn+ nax - b = 0, If ( α1, ,α2,, αn )( are )( the ) roots of equation show that 129 α1- α2 α1- α2 α1- αn = n(α1n - 1 + a)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits Evaluate: lim x→0xx 130  Watch Free Video Solution on Doubtnut

  30. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule ForCENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits 131 Evaluate: (dy/dx)oftanxlogsinx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits Evaluate: lim x→0logcosx 132 x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits 2x- 1 Evaluate: (lim)x 133 →0 1 2 - 1 (1 + x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_L'Hopital' Rule For Evaluating Limits If lim x→aax- xa 134 xx- aa= - 1 and a > 0, then find the value of a.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding Unknowns When Limit Is Give If L = lim x→0sin2x + asinx is finite, then find the value of aandL. 135 x3  Watch Free Video Solution on Doubtnut

  31. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding UnknownsCENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding Unknowns When Limit Is Give tan2x - nsinx x3 If m,n ∈ I0and(lim)x→0 = some integer, then find the value of n and also 136 the value of limit.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding Unknowns When Limit Is Give cos4x + acos2x + b x4 137 If (lim)x is finite, find aandb using expansion formula. →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding Unknowns When Limit Is Give Find the values of aandb in order that = 1 us ∈ gL′Hoπtal′sre. [ ] x(1 + acosx) - bsinx x3 138 ( lim )x→0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding Unknowns When Limit Is Give x3 cos2x - cosx - excosx + ex- 2 139 Find the integral value of n for which lim x→0 is a finite xn nonzero number  Watch Free Video Solution on Doubtnut

  32. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding UnknownsCENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding Unknowns When Limit Is Give aex- b x 140 = 2, then find the value of aandb. If (lim)x →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding Unknowns When Limit Is Give { } x2+ 1 x + 1 141 = 0, then find the value of aandb. If (lim)x - (ax + b) → ∞  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding Unknowns When Limit Is Give 2 x= e3, then find the value of a and b. If lim x→0 1 + ax + bx2 ( ) 142  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Finding Unknowns When Limit Is Give Find the value of α so that lim x→01 3 2 ( ) x2eαx- ex- x = 143  Watch Free Video Solution on Doubtnut

  33. CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical Find the derivative of e√x w.r.t. x using the first principle. 144  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical If f(x) = xtan- 1x, find f′√3 using the first principle. ( ) 145  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical Find the derivative of √4 - x w.r.t. x using the first principle. 146  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical f(x) ( ) -f′(x) [f(x)]2 Using the first principle, prove that d 1 147 = dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical (a + h)2sin-1(a + h) - a2sin-1a h 148 Evaluate ( lim )h →0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical ( ) f(x) = [2x]sin3πxandf′k′ = λkπ( - 1)k (where [.] denotes the greatest integer function 149 and k ∈ N), then find the value of λ .  Watch Free Video Solution on Doubtnut

  34. CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical Differentiate the function with respect to x using the first principle : √sinx 150  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical Differentiate the function with respect to x using the first principle : cos2x 151  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical Differentiate the function with respect to x using the first principle : tan- 1x 152  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical Differentiate the function with respect to x using the first principle : (log)ex 153  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Meaning Of A Derivative AND DERIVATIVES_Geometrical dx(f(x)g(x)) = f(x)d d d 154 Using the first principle, prove that: dx(g(x)) + g(x) dx(f(x))  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions

  35. ( )( )( 1 4 , then find dy ) 1 1 155 . If y = 1 + x 1 + x 1 - x 4 2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions If f(x) = x|x|, then prove that f′(x) = 2|x| 156  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions 0,π ( ) ( ) π 2, π , then find dy 1 - cos2x 1 + cos2x,x ∈ √ . 157 If y = ∪ 2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Derivative Of Some Standard Functions x2 2!+ x3 3!+ + xn a!, then show that dy xn n!= 0 x 158 If y = 1 + 1!+ dx- y +  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Derivatives AND DERIVATIVES_Theorems On Find dy dxfory = xsinxlogx. 159  Watch Free Video Solution on Doubtnut

  36. CENGAGE_MATHS_CALCULUS_LIMITS Derivatives AND DERIVATIVES_Theorems On ex 160 Differentiate y = 1 + sinx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Derivatives AND DERIVATIVES_Theorems On 1 - x2dy ( ) 1 - x 1 + x, prove that √ 161 If y = + y = 0 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Derivatives AND DERIVATIVES_Theorems On ( ) . . . . π 4 162 cos16x, then find f′ . If f(x) = cosx cos2x cos4x cos8x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Derivatives AND DERIVATIVES_Theorems On cos2(a + y) sina If cosy = xcos(a + y), with cosa ≠ ± 1, prove that dy 163 . dx=  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Theorems On

  37. Derivatives x2 y2 1 - y6 1 - x6 √ If √1 - x6+√1 - y6= a x3- y3, then prove that dy ( ) dx= 164  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Derivatives AND DERIVATIVES_Theorems On a - x x - b, thenf ∈ ddy √ If y =√(a - x)(x - b) -(a - b)tan-1 . 165 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy dx for y = sin-1(cosx), where x ∈ (0, 2π). 166  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Differentiate sin-12x√1 - x2 with respect to x if - ( ) 1 < x < 1√2 ="" 1 = √2 < x < 1 ( ) ( ) √2 167 1 ="" (c)="" -1 < x < - √2  Watch Free Video Solution on Doubtnut

  38. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions { , - π < x < π } Find dy 1 - cosx sinx 168 dx for y = tan-1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If y = sin- 1x√1 - x -√x√1 - x2 and 0 < x < 1, then find dy [ ) 169 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy 170 dx for `y=tan^(-1)sqrt((a-x)/(a+x),)-a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions tan- 11 1 + x + x2+ tan-11 x2+ 3x + 3 tan- 11 x2+ 5x + 7 If y = + + upto n terms, then find the value of + 171 y′(0).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy dx for the functions: y = x3exsinx 172  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy x + sinx x + cosx 173 dx for the functions: y =  Watch Free Video Solution on Doubtnut

  39. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( )( 1 + x8 and, )( ) Find the derivative of the function given by f(x) = (1 + x) 1 + x2 1 + x4 174 hence, find f′(1).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) 1 + x2, then find dy 2x 175 If y = sin- 1 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) If y = tan- 13x - x3 1 √3, then find dy 1 176 1 - 3x2, < x < dx √3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions 177

  40. ( ) 1 ; 0 < x < √2 , ="" then="" find="" dy ( ) If y = sec-1 dx 2x2- 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions {( )} 3 Find dy x - 2 x + 2 178 4 dx if y = log ex  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) + sin-1√x - 1 ( ) dx if y = sec-1√x + 1 Find dy 179 √x - 1 √x + 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) + tan-12 + 3x ( ) Find dy 4x 180 dx if y = tan- 1 1 + 5x2 3 - 2x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of

  41. Inverse Trigonometric Functions 181 tan- 1(4x) 1 + 5x2 +tan-1(2 + 3x) 3 - 2x Find dy dx if y =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) dxify = tan-1√1 + x2- 1 F i n ddy 182 , where x ≠ 0 x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) acosx - bsinx bcosx + asinx , where -π 2< x < πand a 183 y = tan-1 btanx > - 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) y = tan-1√1 + x2+√1 - x2 √1 + x2-√1 - x2  Watch Free Video Solution on Doubtnut 184 ,where `-1 CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) ( ) x 1 185 y = sin-1 + cos-1 √1 + x2 √1 + x2  Watch Free Video Solution on Doubtnut

  42. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions tan-13a2x - x3 ( ( ) 186 y = ) a a2- 3x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) 5x + 12√1 - x2 13 187 y = sin-1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) x y = tan-1 188 1 +√1 - x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions y = sin-1√x - ax -√a - ax [ ] 189  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of

  43. Inverse Trigonometric Functions 190 Find the sum of the series 1 + 2x + 3x2+ (n - 1)xn- 2 using differentiation.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy ( ) dx for y = sin x2+ 1. 191  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions log sin √{ ( )} x2 3- 1 , then find dy 192 . If y = dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ( ) 193 Differentiate the function f(x) = sec tan√x with respect to x.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions ( ) Find dy dxf or y = log x +√a2+ x2. 194

  44.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions → R be a one-one onto differentiable function, such that f(2) = 1andf′(2) = 3. The (( ( Let f:R ))) 195 d f-1(x) find the value of dx x=1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy dx for the function: y = sin-1√(1 - x) + cos-1√x 196  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy 197 dx for the function: y =√sin√x  Watch Free Video Solution on Doubtnut

  45. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy dx for the function: y = esinx^ 2 198  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy √ sin√ex 199 dx for the function: y = log  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy dx for the function: y = asin-1x^ 2 200  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions Find dy π 3 1 + sinx 1 - s ∈ gx, wherex = √ dx for the function: y = (log)e 201  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions 1 + x2n, then find dy )( ) ( 1 + x4 )( If y = (1 + x) 1 + x2 202 dxatx = 0.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If xy= ex-y, Prove that dy logx 203 dx= (1 + logx)2  Watch Free Video Solution on Doubtnut

  46. CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Inverse Trigonometric Functions If x√1 + y + y√1 + x = 0, prove that dy 1 204 dx= - (x + 1)2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions If xy + y2= tanx + y, thenf ∈ ddy 205 . dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions If √x +√y = 4, thenf ∈ ddx 206 dyaty = 1.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions ( ) = 2tan- 1y , show that dy x + y x - y If log x2+ y2 ( ) 207 dx= x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions If x3+ y3+ 3axy = 0, f ∈ ddy 208 . dx

  47.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions If y =√sinx + y, thenf ∈ ddy . 209 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions ( ( )) If y = btan- 1x a+ tan-1y ,f ∈ ddy 210 x dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions sinx +√sinx +√sinx + → ∞, provethatdy √ cosx 2y - 1 211 If y = dx=  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Implicit Functions y2- x y +√x +√y + ∞, thenprovethatdy √ √ 212 If y = dx= x + 2y3- 2xy - 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form

  48. Find dy 213 if x = a(θ - sinθ)andy = a(1 - cosθ). dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form If x = asec3θ and y = atan3θ, find dy π 3 . dx at θ = 214  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form . If(dy) dx = 2andx = 3a = 0, then find the value of dx Let y = x3- 8x + 7andx = f(t) 215 dta = 0.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form 1 - t2 1 + t2, thenf ∈ ddy 2t 216 If x = 1 + t2, y = dxa = 2.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form If x = acos3θ,y = bsin3θ,f ∈ dd3y 217 dx3 at θ = 0.

  49.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form acos-1ta > 0and-1 < t < 1 , show that dy √ y x, asin-1t, y = √ If x = dx= - 218  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS AND DERIVATIVES_Differentiation Of Functions In Parametric Form If x = a cost +1 ( 2logtan2t and y = asint then find dy ) π 4 219 dx at t =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Using Logarithm AND DERIVATIVES_Differentiation If xmyn= (x + y)m+n,provethatdy y x 220 . dx=  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Using Logarithm AND DERIVATIVES_Differentiation Differentiate (logx)cosx with respect to x. 221  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Using Logarithm AND DERIVATIVES_Differentiation

  50. ( ) π 4 222 If f(x) = |x||sinx|, then find f′ -  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Using Logarithm AND DERIVATIVES_Differentiation If y = xxxx... ∞ , find dy 223 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Using Logarithm AND DERIVATIVES_Differentiation 3 Find the derivative of √x(x + 4) 2 224 4 (4x - 3) 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Using Logarithm AND DERIVATIVES_Differentiation 4x3 1 + x4+ ∞ = 1 2x 1 225 If x < 1, provethat 1 + x+ 1 + x2+ 1 - x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Using Logarithm AND DERIVATIVES_Differentiation Find dy dxf or y = xx. 226  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_LIMITS Using Logarithm AND DERIVATIVES_Differentiation Differentiate (xcosx)x with respect to x. 227  Watch Free Video Solution on Doubtnut

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