chapter2(5) - r x is defined for all x , then 1 lim = r x x...

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MATH1010: Chapter 2 cont… 1 LIMITS AND DERIVATIVES cont… Limits at Infinity; Horizontal Asymptotes (Section 2.6, pg. 135) Recall: Previously, we talked about infinite limits and vertical asymptotes. Horizontal asymptotes, on the contrary, are based on the behaviour as x and −∞ x . x f(x) 10 1.540541 100 1.950401 1000 1.995004 10000 1.9995 100000 1.99995 1000000 1.999995 Definition: Let f be a function defined on some interval ) , ( a . Then L x f x = ) ( lim means that the values of ) ( x f can be made arbitrarily close to L by taking x sufficiently large. [Similarly, we can define L x f x = ) ( lim ] Definition: The line L y = is called a horizontal asymptote of the curve ) ( x f y = if either L x f x = ) ( lim or L x f x = ) ( lim . Graphical Examples:
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MATH1010: Chapter 2 cont… 2 Now, suppose we aren’t given the graph of a function, and let’s try to compute the limits at infinity. Example: 2 1 ) ( x x f = Question: Does ) ( lim x f x and ) ( lim x f x always approach a particular value L ? Theorem: If 0 > r is a rational number, then 0 1 lim = r x x and if 0 > r is a rational number such that
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Unformatted text preview: r x is defined for all x , then 1 lim = r x x Application: Suppose that the diameter of an animals pupils is given by ) ( x f mm, where x is the intensity of light on the pupils. If 15 4 90 160 ) ( 4 . 4 . + + = x x x f , find the diameter of the pupils with maximum light. [Source: Calculus: Concepts & Connections by R. Smith and R. Minton, 2006] Example: 9 5 2 7 lim 2 2 + x x x x MATH1010: Chapter 2 cont 3 Note: Our strategy for infinite limits of rational functions is to divide by the largest power in the denominator ! Example: 7 3 1 5 lim 2 x x x Example: 6 3 6 2 4 3 8 7 5 2 lim x x x x x + Example: 4 2 5 lim 2 + x x x What happens if we instead consider the limit as x ? i.e. 4 2 5 lim 2 + x x x Example: 2 3 6 3 lim 2 2 + x x x Example: 6 8 lim x x x...
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chapter2(5) - r x is defined for all x , then 1 lim = r x x...

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