When we find the limit of a function *f*(*x*) as *x* goes to infinity, we're answering the question "What value is *f*(*x*) approaching as *x* gets bigger and bigger and bigger...?''

A rockin' example of this is

As *x* gets bigger and bigger, gets closer and closer to zero. We can get a feel for this by making a table:

Since is approaching 0 as *x* gets larger,

Similarly,

This makes sense on the graph, since the bigger *x* gets, or the more negative *x* gets, the closer *y* gets to zero:

Graph of :

If we had some other constant in the numerator besides 1, the limit would still be 0. We would still be dividing a constant by larger and larger numbers. Therefore,

In fact, whenever we have a rational function *f*(*x*) where the degree of the numerator is less than the degree of the denominator,

Sometimes the limit of a function as *x* goes to ∞ is undefined.

## Practice:

If *f*(*x*) = sin(*x*), find | |

This limit is not defined. Look at the graph of *f*(*x*): As *x* gets bigger and bigger, *f* will sometimes be 1 and sometimes *-1*. There is no one value that *f*(*x*) is approaching - it's bouncing around all over the place. This limit does not exist.
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If *f*(*x*) = *x* + 1, find | |

As *x* gets bigger, *f*(*x*) keeps getting bigger, too. There is no number that *f*(*x*) is approaching, because *f*(*x*) keeps growing without bound. This limit does not exist. We could also say since *f*(*x*) is forever getting larger and is unbounded. If we have a rational function where the degree of the numerator is smaller than the degree of the denominator, what happens to the value of that rational function as *x* approaches ∞? | |

Use a table to estimate | |

What happens as *x* gets larger and larger? It looks like the quotient is approaching 0, so we'll say The same will be true for any rational function where the degree of the numerator is smaller than the degree of the denominator. We know because we're dividing 1 by larger and larger things as *x* approaches infinity. If we have a rational function where the degree of *p*(*x*) is smaller than the degree of *q*(*x*), *q* will get larger "faster" than *p* will, and the fraction will approach 0. We'll talk more about this in a bit, and include more pictures, when we compare functions and their limits at infinity more generally. | |

Find the limit, if it exists.

Answer

0 since

Find the limit, if it exists.

Answer

-∞ or "does not exist.'' As *x* gets more and more negative, so does -x^{2}.

Find the limit, if it exists.

- where

Hint

2*k* < *x* < 2*k* + 1 means *x* is greater than an even integer and less than the next integer. For example, we could have 2*x* < 9

Answer

The limit does not exist, since *f*(*x*) can't decide whether it's trying to approach 1 or 0.

Find the limit, if it exists.

Answer

∞ or "does not exist.'' As *x* gets larger, so does 2^{x}.

Find the limit, if it exists.

Answer

as *x* gets larger and larger, approaches 2.

Find the limit, if it exists.

Answer

the limit is 0 since the degree of the numerator is smaller than the degree of the denominator.