From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!

# 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.