Asymptotes and Limits at Infinity

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We say that a function has a limit as approaches if for every positive tolerance , we can always find a cutoff such that if , then approximates with an error less than .

That is,

if , then .

In this case, we write
We can also define limits at in a similar manner. In particular, we say that a function f has a limit as approaches if for every positive tolerance , we can always find a cutoff such that if , then approximates with an error less than .

That is

if , then

In this case we write

Note: The usual rules for the arithmetic of limits hold for limits at . In fact, the Squeeze Theorem > Squeeze Theorem for Limits at $infty$ also holds with the proper modifications.