Limits at or with infinity
[math]\displaystyle{ \lim_{x \ \to \ \infty} x^2 = \infty }[/math] because the function can grow indefinitely. [math]\displaystyle{ \lim_{x \ \to \ \infty} \frac{1}{x} = 0 }[/math] because we are dividing a number by something very large, or in infinitely many small parts. Is there a rigorous way to prove that our intuition is correct in both cases? Yes, there is. This idea is pretty abstract because the whole concept is "there is a number that is very large, then we can add one and make it even larger and continue this process indefinitely". The other side of the same coin is to divide a number and repeat it indefinitely. We know that to divide anything in half yields smaller parts. Doing it again and again would come to an end at the atomic and then, subatomic level. But mathematics allow us to go beyond the smallest atom and there is the abstraction, how can something be smaller than a subatomic particle?
The definition of a limit when we have infinity is really the same for calculating it at a given, finite, point. We begin by stating:
[math]\displaystyle{ f(x) \gt M }[/math]. M is a large number, the largest number that one could think of. We are stating that no matter how large M is, [math]\displaystyle{ f(x) }[/math] is still larger than that. Then,
[math]\displaystyle{ 0 \lt |x - a| \lt \delta }[/math]. The distance between two consecutive points on the function is positive. [math]\displaystyle{ \delta }[/math] is greater than that distance.
(If you didn't quite grasp the concept, try a large number, such as 99 or 1000. Calculate the function for that number, try an even bigger number and do it again. This is what the letters epsilon and delta mean.)
Now for the side limits:
If [math]\displaystyle{ f(x) \gt M }[/math] whenever [math]\displaystyle{ 0 \lt x - a \lt \delta }[/math], we write
[math]\displaystyle{ \lim_{x \ \to \ a^{+}} = +\infty }[/math]
[math]\displaystyle{ \lim_{x \ \to \ a^{-}} = +\infty }[/math]
If both sides converge to the same limit, it does exist and is infinite itself.
