G. Limits at Infinity

We occasionally want to know what happens to some quantity when a variable gets very large or “goes to infinity”.

Example: Limit at Infinity.

What happens to the function cos(1/x) as x goes to infinity? It seems clear that as xx gets larger and larger, 1/x gets closer and closer to zero, so cos(1/x) should be getting closer and closer to cos(0)=1.

As with ordinary limits, this concept of “limit at infinity” can be made precise. Roughly, we want limx→∞f(x)=L to mean that we can make f(x) as close as we want to LL by making xx large enough.

Limit at Infinity.

In general, we write

limx→∞f(x)=L

if f(x) can be made arbitrarily close to L by taking x large enough. If this limits exists, we say that the function f has the limit L as x increases without bound.

Similarly, we write

limx→−∞f(x)=M

if f(x) can be made arbitrarily close to M by taking x to be negative and sufficiently large in absolute value. If this limit exists, we say that the function ff has the limit L as x decreases without bound.

Example: