Limits in the infinite

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:

$$\lim_{x \to -\infty}{f(x)} = \lim_{x \to -\infty}{x^2-1}=(-\infty)^2-1=\infty$$

$$\lim_{x \to +\infty}{f(x)} = \lim_{x \to +\infty}{x^2-1}=\infty^2-1=\infty$$