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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 one-sided 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 one-sided 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 ( y-axis) 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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