Section 2.5 - Section 2.5: Infinite Limits Formal...

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Section 2.5: Infinite Limits Formal Definition: Infinite Limits Let f be a function that is defined at every real number in some open interval containing c (except possibly at c itself). The statement ( ) lim x c f x = ∞ means that for each M>0, there exists a 0 δ > such that ( ) x c f x M - < > . Similarly, the statement ( ) lim x c f x = -∞ means that for each N>0, there exists a 0 > such that ( ) x c f x N - < < . ( ) lim x c f x = ∞ does NOT mean that the limit exists! It simply says that the limit does not exist due to unbounded behavior at x = c. Definition: Vertical Asymptote A function ( ) y f x = has a vertical asymptote at x c = iff ( ) lim x c f x - = ±∞ or ( ) lim x c f x + = ±∞ . Finding Vertical Asymptotes of Quotient Functions Let ( ) ( ) ( ) f x h x g x = . If ( ) ( ) 0 and 0, g c f c = then x c = is a vertical asymptote of ( ) h x . If ( ) ( ) 0 and 0, g c f c = = then ( ) h x has a hole in the graph at x c = . Finding Vertical Asymptotes of Rational Functions
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Section 2.5 - Section 2.5: Infinite Limits Formal...

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