Chapter 2 | Limits
149
Figure 2.20 The function f ( x ) =1/( x − a ) n has infinite limits at a .
Theorem 2.3: Infinite Limits from Positive Integers If n is a positive even integer, then lim x → a 1 ( x − a ) n
=+∞.
If n is a positive odd integer, then
1 ( x − a ) n
lim x → a +
=+∞
and
1 ( x − a ) n
lim x → a −
=−∞.
We should also point out that in the graphs of f ( x ) =1/( x − a ) n , points on the graph having x -coordinates very near to a are very close to the vertical line x = a . That is, as x approaches a , the points on the graph of f ( x ) are closer to the line x = a . The line x = a is called a vertical asymptote of the graph. We formally define a vertical asymptote as follows: Definition Let f ( x ) be a function. If any of the following conditions hold, then the line x = a is a vertical asymptote of f ( x ). lim x → a − f ( x ) = +∞or−∞ lim x → a + f ( x ) = +∞or−∞ or lim x → a f ( x ) = +∞or−∞
Example 2.10
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