Infinity - LIMITS INVOLVING INFINITY When a function has an...

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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 0 + 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 0 1 x 2 - 1 x . Solution Since x 2 approaches 0 while 1 - x approaches 1 as x approaches 0, we can see that 1 - x x 2
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