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# Asymptote of a Function

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Tool to find the equations of the asymptotes (horizontal, vertical, oblique) of a function. The asymptotes are lines that tend (similar to a tangent) to function towards infinity.
Summary

## Asymptotes of a Function Calculator

### How to find an horizontal asymptote?

A function $f(x)$ has an horizontal asymptote $y = a$ if

$$\lim\limits_{x \rightarrow +\infty} f(x)=a \mbox{ or } \lim\limits_{x \rightarrow -\infty} f(x)=a \mbox{ (or both)}$$

Example: $1/x$ has for asymtote $y=0$ because $\lim\limits_{x \rightarrow \infty} 1/x = 0$

There can not be more than 2 horizontal asymptotes.

### How to find a vertical asymptote?

A function $f(x)$ has a vertical asymptote $x = a$ if it admits an infinite limit in $a$ ($f$ tends to infinity).

$$\lim\limits_{x \rightarrow \pm a} f(x)=\pm \infty$$

Example: $1/x$ has for asymtote $x=0$ because $\lim\limits_{x \rightarrow 0} 1/x = \infty$

Generally, the function is not defined in $a$, it is necessary to analyze the domain of the function to find potential asymptotes.

There may be an infinite number of vertical asymptotes.

### How to find a slant/oblique asymptote?

A function $f(x)$ has a slant asymptote $g(x)=ax+b$ when

$$\lim\limits_{x \rightarrow \pm \infty} \left( f(x)-g(x)= 0 \right)$$

Computation of slant asymptote may be simplified by calculating this limit :

$$\lim\limits_{x \rightarrow \pm \infty} \left( \frac{f(x)}{g(x)} = 1 \right)$$

### How to find a non-linear asymptote?

A function $f(x)$ has a non-linear asymptote $g(x)$ when

$$\lim\limits_{x \rightarrow \pm \infty} \left( f(x)-g(x)= 0 \right)$$

The method is the same as the oblique asymptote calculation.

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