LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS
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1 LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS
2 RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or negative). For a rational function r x = p x q x, this occurs when p a 0 and q a = 0. The following properties will hold true when performing operations with functions which may have asymptotes: Suppose a and L are real numbers, and let f and g be functions such that f x = and g(x) = L x a x a Sum and Difference f x ± g x x a = Product f x g x =, L > 0 x a f x g x =, L < 0 Quotient x a g x x a f x = 0
3 LIMITS AT INFINITY Consider the function below, and let s ask what happens to the function as x increases to positive infinity or negative infinity. f x = x2 x We can analyze this end behavior of the function using the tabular method. Set up an x-y table and choose values of x that get larger and larger. It s usually easier to use powers of ten to make the arithmetic easier. What we re trying to find is: f(x) and f(x) x
4 LIMITS AT INFINITY f x = x2 x The graph of f x shows that the function approaches 1 when x gets larger and larger.
5 LIMITS AT INFINITY Let f(x) be defined on some interval a,. We say that f(x) has it L as x approaches if the values of f(x) can be made arbitrarily close to L by choosing x sufficiently large. f(x) = L We have similar definitions as x For Consideration: What are the its of the function f x = 1/x as x goes to positive and negative infinity?
6 HORIZONTAL ASYMPTOTES We call the line y = L a horizontal asymptote of the graph of f(x) if either f x = L x f x = L These its to infinity have many of the same properties as its discussed in previous sections. For instance: For functions f and g with its as follows f x = L g x = M f x ± g x = L ± M f x g x = LM
7 HORIZONTAL ASYMPTOTES How many horizontal asymptotes can a graph have? Consider the following functions: f x = 1/x g x = 1 e x h x = tan 1 x
8 INFINITE LIMITS AT INFINITY Let f(x) be defined on some interval a,. We say that f(x) has it as x approaches if the values of f(x) can be made as arbitrarily large as we want by choosing x sufficiently large. f(x) = We have similar definitions for f x =, f x =, and x x Try finding the following its at infinity: x x x2 f x = x x x2
9 EVALUATING LIMITS AT INFINITY If r is a positive rational number and c is any real number, then: c x r = 0 and x c x r = 0 Examples: Evaluate the following its: 5 3 x x 5 + 1
10 EVALUATING LIMITS AT INFINITY For rational functions, we can find horizontal asymptotes by essentially dividing the numerator and denominator by the highest power of x. Let s compare the following three examples: 2x + 3 3x x x x x Each of these its produces an indeterminate form /. For the first two its, we can divide both the numerator and denominator by x 2, and for the third, divide both by x 3
11 HORIZONTAL ASYMPTOTES FOR RATIONAL FUNCTIONS If f x = p x q x, where p(x) is a polynomial of degree m and q(x) is a polynomial of degree n, then 1. If m < n, then f x has a horizontal asymptote at y = If m = n, then f(x) has an asymptote at y = a, where a b and b are the leading coefficients of p and q, respectively. 3. If m > n, then f(x) has no horizontal asymptote. Examples: Find horizontal asymptotes for the following rational functions f x = 6x2 + 1 x 4 g x = x2 + 5x 36 4x 2 + x h x = x3 + 6x 2x 1
12 FUNCTIONS WITH TWO HORIZONTAL ASYMPTOTES Occasionally, we run into functions that involve division of radicals. In these instances, we take advantage of the fact that x 2 has two different values. Consider the following examples: 3x 4x 2 8 x 3x 4x 2 8 For the first example, we let x 2 = x, and divide both the numerator and denominator by x. For the second, we let x 2 = x and do the same thing.
13 ADDITIONAL EXAMPLES Consider the following its at infinity: ex x ex e x e 1/x (these are better seen by graphing; use a graphing utility!!) 3x(2x 2 + 1) 6 x 2 2x 3 3e x 2e x 3 sin x x (Hint: Try Squeeze Theorem)
14 CLASSWORK & HOMEWORK MATH JOURNAL: Summarize what you learned today CLASSWORK: HORIZONTAL ASYMPTOTES Find an example of a function for each of the following conditions, and describe the its of each at positive and negative infinity: a) A rational function with no horizontal asymptotes b) A function with one horizontal asymptote and two vertical asymptotes c) A function with a horizontal asymptote at y = 1 2 d) A function with two horizontal asymptotes Homework: Pg. 205, #1-38
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