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?

Example

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:

$lim_{x \to + \infty} f (x) and lim_{x \to - \infty} f (x)$

Example

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:

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

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