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A2 – Determining Limits of Functions Graphically. IB Math HL&SL - Santowski. (A) Review.

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a review
(A) Review
  • To understand how functions can have limits at any given x value, we simply understand the idea that we are approaching a given x value from both the left side and right side and then simply making observations about how the function values (the y values) are behaving.
  • If the function value (y value) is the same as we approach the x value from both the right and the left, then we say that the function does indeed have a limiting value at that given x value.
b properties of limits limit laws
(B) Properties of Limits - Limit Laws
  • The limit of a constant function is the constant
  • The limit of a sum is the sum of the limits
  • The limit of a difference is the difference of the limits
  • The limit of a constant times a function is the constant times the limit of the function
  • The limit of a product is the product of the limits
  • The limit of a quotient is the quotient of the limits (if the limit of the denominator is not 0)
  • The limit of a power is the power of the limit
  • The limit of a root is the root of the limit
c limits limit laws and graphs5
From the graph on this or the previous page, determine the following limits:

(1) lim x-2 [f(x) + g(x)]

(2) lim x-2 [(f(x))2 - g(x)]

(3) lim x-2 [f(x) × g(x)]

(4) lim x-2 [f(x) ÷ g(x)]

(5) lim x1 [f(x) + 5g(x)]

(6) lim x1 [ ½f(x) × (g(x))3]

(7) lim x2 [f(x) ÷ g(x)]

(8) lim x2 [g(x) ÷ f(x)]

(9) lim x3 [f(x) ÷ g(x)]

(C) Limits, Limit Laws, and Graphs
d limits and graphs7
(A) Find function values at the following:

(i) f(-10) =

(ii) f(-4) =

(iii) f(2) =

(iv) f(7) =

(D) Limits and Graphs
d limits and graphs8
(D) Limits and Graphs
  • (B) Find the limit of the function f(x) from the previous slide at the following values:
  • (i) x = -10- and x = -10+ and so therefore, the limit of the function at x = -10 is
  • (ii) x = -6- and x = -6+ and so therefore, the limit of the function at x = -6 is
  • (iii) x = -4- and x = -4+ and so therefore, the limit of the function at x = -4 is
  • (iv) x = 0- and x = 0+ and so therefore, the limit of the function at x = 0 is
  • (v) x = 2- and x = 2+ and so therefore, the limit of the function at x = 2 is
  • (vi) x = 4- and x = 4+ and so therefore, the limit of the function at x = 4 is
  • (vii) x = 6- and x = 6+ and so therefore, the limit of the function at x = 6 is
  • (i) x = 7- and x = 7+ and so therefore, the limit of the function at x = 7 is
  • (i) x = 10- and x = 10+ and so therefore, the limit of the function at x = 10 is
e graphs gdc and limits
(E) Graphs, GDC, and Limits
  • Now use a GDC to help you determine the following limits:
  • (i)
  • (ii)
  • (iii)
f one sided and two sided limits
(F) One-Sided and Two-Sided Limits
  • In considering several of our previous examples, we see the idea of one and two sided limits.
  • A one sided limit can be a left handed limit notated as which means we approach x = a from the left (or negative) side
  • We also have right handed limits which are notated as which means we approach x = a from the right (or positive) side
  • We can make use of the left and right handed limits and now define conditions under which we say a function does not have a limiting y value at a given x value  by again considering our various examples above, we can see that some of our functions do not have a limiting y value because as we approach the x value from the right and from the left, we do not reach the same limiting y value.
  • Therefore, if then does not exist.
g internet links
(G) Internet Links
  • On-line quiz from Visual Calculus on Limit Theorems
  • Tutorial and Quiz from Harvey Mudd College on Computing Limits
  • Limits - One-Sided Limits - from Paul Dawkins at Lamar U
  • Visual Calculus - Numerical Exploration of Limits
h homework
(H) Homework
  • Stewart, 1989, Calculus – A First Course, Chap 1.2, p18, Q1 & Chap 1.3, p27, Q1-3,4,6,8,9