Calculus/Infinite Limits

Informal infinite limits

Limits can also involve looking at what happens to $f(x)$ as x gets very big. For example, consider the function 1/x. As x gets very big, 1/x gets very small. 1/x gets closer and closer to zero, the bigger x gets. Now without limits it is very difficult to talk about this fact, because 1/x never actually gets to zero. But the language of limits exists precisely to let us talk about the behavior of a function as it approaches something, without caring about the fact that it will never get there. In this case, however, we have the same problem as before; how big does x have to be to be sure that $f(x)$ is really going towards 0?

In this case, we can say that the bigger x gets, the closer f(x) should get to 0. Really, this means that however close we want f(x) to get to 0, we can find an x big enough so f(x) is that close. We write this in a similar way to regular limits:

$\underset{x\to \mathrm{\infty }}{\lim }\frac{1}{x}=0.$

Which is said "The limit, as x approaches infinity, equals 0," or "as x approaches infinity, the function approaches 0". Notice, however, that infinity is not a number; it's just shorthand for saying "no matter how big."

Examples

Find $\underset{x\to \mathrm{\infty }}{\lim }\frac{x-5}{x-3}$