Continuity#2

# Continuity#2 - MATH1010 Chapter 2 cont 1 LIMITS AND...

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MATH1010: Chapter 2 cont… 1 LIMITS AND DERIVATIVES cont… Limits at Infinity; Horizontal Asymptotes (Section 2.6, pg. 130) 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 r x is defined for all x , then 0 1 lim r x
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Continuity#2 - MATH1010 Chapter 2 cont 1 LIMITS AND...

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