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Unformatted text preview: LIMITS INVOLVING INFINITY When a function has an infinite discontinuity, we say the limit of the function is infinity. Symbolically, we write lim x → a f ( x ) = ∞ to indicate that the values of f ( x ) becomes larger and larger (arbitrarily large) as x approaches a . Similarly, if the values of f ( x ) can be made as large negative as we wish for all values of x sufficiently close to a , we write lim x → a f ( x ) =∞ . Similar definitions can be made for onesided limits. Note that, even if we have ∞ as a limit, it does not mean that the limit exist. The line x = a is called a vertical asymptote of the curve y = f ( x ) if at least one of onesided limits at a is infinity or negative infinity. Example 1. Find an equation of a vertical asymptote of the curve y = ln x . Solution Since lim x → + ln x =∞ , the line x = 0 ( yaxis) is a vertical asymptote. Limit laws do not work well with infinite limits. For example, ∞ + ∞ = ∞ and ∞ · ∞ = ∞ , but ∞  ∞ cannot be defined. Example 2. Find lim x → 1 x 2 1 x ....
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
 Fall '09
 BenedictGross
 Continuity, Limits

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