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Evaluating Limits Analytically

Evaluating Limits Analytically. Section 1.3. After this lesson, you will be able to:. evaluate a limit using the properties of limits develop and use a strategy for finding limits evaluate a limit using dividing out and rationalizing techniques. Limits  Analytically.

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Evaluating Limits Analytically

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  1. Evaluating Limits Analytically Section 1.3

  2. After this lesson, you will be able to: • evaluate a limit using the properties of limits • develop and use a strategy for finding limits • evaluate a limit using dividing out and rationalizing techniques

  3. Limits Analytically In the previous lesson, you learned how to find limits numerically and graphically. In this lesson you will be shown how to find them analytically…using algebra or calculus.

  4. ____ ____ ____ Let Let Let Theorem 1.1 Some Basic Limits Let b and c be real numbers and let n be a positive integer. Examples Think of it graphically y scale was adjusted to fit As x approaches 5, f(x) approaches 125 As x approaches 3, f(x) approaches 4 As x approaches 2, f(x) approaches 2

  5. Direct Substitution • Some limits can be evaluated by direct substitution for x. • Direct substitution works on continuous functions. • Continuous functions do NOT have any holes, breaks or gaps. Note:Direct substitution is valid for all polynomial functions and rational functions whose denominators are not zero.

  6. and Theorem 1.2 Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: Scalar multiple: Sum or difference: Product: Quotient: Power:

  7. Limit of a Polynomial Function Example: Since a polynomial function is a continuous function, then we know the limit from the right and left of any number will be the same. Thus, we may use direct substitution.

  8. Limit of a Rational Function Make sure the denominator doesn’t = 0 ! Example: If the denominator had been 0, we would not have been able to use direct substitution.

  9. Theorem 1.4 The Limit of a Function Involving a Radical Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is even. So we can use direct substitution again, as long as c is in the domain of the radical function.

  10. Theorem 1.5 The Limit of a Composite Function If f and g are functions such that lim g(x) = L and lim f(x) = f(L), then

  11. Limit of a Composite Function-part a Example: Given and , find a) First find Direct substitution works here

  12. b) Then find Therefore, Limit of a Composite Function -part b Direct substitution works here, too. *********************************************

  13. Limits of Trig Functions If c is in the domain of the given trig function, then

  14. Limits of Trig Functions Examples:

  15. Limits of Trig Functions Examples:

  16. Finding Limits • Try Direct Substitution • If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to divide out common factors or to rationalize the numerator so that direct substitution works. • Use a graph or table to reinforce your result.

  17. Factor the denominator Direct substitution at this point will give you 0 in the denominator: Now direct substitution will work Example 1- Factoring Example* : Graph on your calculator and use the table to check your result

  18. Direct substitution results in the indeterminate form 0/0. Try factoring. Example 2- Factoring Example:

  19. Sum of cubes Not factorable Direct substitution results in 0 in the denominator. Try factoring. None of the factors can be divided out, so direct substitution still won’t work. The limit DNE. Verify the result on your calculator. The limits from the right and left do not equal each other, thus the limit DNE. Observe how the right limit goes to off to positive infinity and the left limit goes to negative infinity. Example: Limit DNE Example*:

  20. Indeterminate Form Example 1- Rationalizing Technique Example*: First, we will try direct substitution: Plan: Rationalize the numerator to come up with a related function that is defined at x = 0.

  21. Multiply the numerator and denominator by the conjugateof the numerator. Note: It was convenient NOT to distribute in the denominator, but you did need to FOIL in the numerator. Now direct substitution will work Example 1- Rationalizing Technique Example*: Go ahead and graph to verify.

  22. Example 2- Rationalizing Technique Example:

  23. Two Very Important Trig Limits (A star will indicate the need to memorize!!!)

  24. Note: Before you decide to even use a special trig limit, make sure that direct substitution won’t work. In this case, direct substitution won’t work, so let’s try to get this to look like one of those special trig limits. This 5 is a constant and can be pulled out in front of the limit. Equals 1 Example 1- Using Trig Limits Example*: Now, the 5x is like the heart. You will need the bottom to also be 5x in order to use the trig limit. So, multiply the top and bottom by 5. You won’t have changed the fraction. Watch how to do it.

  25. Try to get into the form: Example 2

  26. Example 3

  27. Get into the form: Example 4

  28. Example 5

  29. Homework Section 1.3: page 67 #1, 5-39 odd, 49-61 odd, 67-77 odd

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