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Functions, Graphs, and Limits

Functions, Graphs, and Limits

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.

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