# Limits of Functions at Infinity

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. Take *f*(*x*) = sin *x*. As *x* goes to infinity, sin *x* just keeps bouncing between 0 and 1 without really ever honing in one number. That limit doesn't exist.