Considering $$f(x)$$ we can ask what happens to $$f(x)$$ when we make $$x$$ very big. In other words, where is $$f(x)$$ going when $$x$$ tends to infinity?
For example, the function $$f(x)=1$$ is constant and its value is always $$1$$. Consequently, its limit when $$x$$ tends to infinity is $$1$$, but the function $$f(x)=x$$ however, tends to infinity when $$x$$ tends to infinity.
The operation of looking for the limit when x tends to infinity of a function is denoted as:
$$$\lim_{x \to \infty}{f(x)}$$$
We must also think that we can make the limit of a function when x becomes very big or when x is very negative. Therefore, we can define the limits of $$f(x)$$ when $$x$$ tends to plus infinity and to minus infinity:
$$$\lim_{x \to +\infty}{f(x)} \ \text{and} \ \lim_{x \to -\infty}{f(x)}$$$
Let's take the function $$f(x)=x^2-1$$.
If we compute its limit when $$x$$ tends to plus and minus infinity we arrive at:
$$$\lim_{x \to -\infty}{f(x)} = \lim_{x \to -\infty}{x^2-1}=(-\infty)^2-1=\infty$$$
$$$\lim_{x \to +\infty}{f(x)} = \lim_{x \to +\infty}{x^2-1}=\infty^2-1=\infty$$$