We define an **asymptote as a straight line that can be horizontal, vertical or obliquous that goes closer and closer to a curve which is the graphic of a given function**.

These asymptotes usually appear if there are points where the function is not defined.

Let's see an example, since it will make it easier to understand.

Let's take the function $$f(x)=\dfrac{1}{x}$$. It is clear that when $$x=0$$ the function is not defined. It is exactly here where the asymptote appears. Let's see the graphic representation of this function:

When we go closer and closer to $$x=0$$ from the right we can observe that the function tends to infinity, increasingly approaching the straight line $$x=0$$. The same happens on the other side, but going to minus infinity when approaching the line $$x=0$$. We say that the line $$x=0$$ is an asymptote.

Let's see a more exact definition of asymptote of a function $$f(x)$$:

## Vertical Asymptote

We will say that the straight line $$x=a$$ (where $$a$$ is a number) is a **vertical asymptote** if there exist some of these two limits:

- $$\displaystyle \lim_{x\rightarrow a^-}f(x)=\pm\infty$$
- $$\displaystyle \lim_{x\rightarrow a^+}f(x)=\pm\infty$$

The function $$f(x)=\dfrac{1}{1+x}$$ has a vertical asymptote in $$x=-1$$, since the limit $$\displaystyle \lim_{x\rightarrow -1^+}f(x)=\pm\infty$$ exists and therefore the asymptote also does.

## Horizontal Asymptote

If the following limit exists:

$$$\displaystyle \lim_{x\rightarrow \pm\infty}f(x)=a$$$

where $$a$$ is a finite value, then we will say that the line $$y=a$$ is an **horizontal asymptote**.

The function $$f(x)=e^x$$ has an horizontal asymptote in $$y=0$$ since:

$$$\displaystyle \lim_{x\rightarrow -\infty}e^x=0$$$

## Obliquous Asymptote

If the following limits exist and are finite:

- $$\displaystyle \lim_{x\rightarrow \infty}\frac{f(x)}{x}=m$$
- $$\displaystyle \lim_{x\rightarrow \infty}(f(x)-mx)=b$$

Then we will say that an **obliquous asymptote** exists and the line of the asymptote is given by the equation $$y=mx+b$$.

The obliquous asymptotes only exist in rational functions (division of polynomials) where the polynomial of the nominator has a greater degree than the denominator.

The function $$f(x)=\dfrac{x^2+1}{x}$$ has an obliquous asymptote since the following limits exist:

$$$\displaystyle m=\lim_{x\rightarrow \infty}\frac{f(x)}{x}=\lim_{x\rightarrow \infty}\frac{x^2+1}{x^2}=1$$$

$$$\displaystyle b=\lim_{x\rightarrow \infty}(f(x)-mx)=\lim_{x\rightarrow \infty} \Big( \frac{x^2+1}{x}-\frac{x^2}{x}\Big)=\lim_{x\rightarrow \infty}\Big( \frac{1}{x} \Big) =0$$$

so the line $$y=x$$ is the obliquous asymptote.

(we observe the obliquous asymptote in green, the straight line being $$y = x$$).