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Asymptote of a Function

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.

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Asymptote of a Function -

Tag(s) : Mathematics

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# Asymptote of a Function

## Asymptotes of a Function Calculator

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.

### 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 $$a$$.

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

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 an oblique asymptote?

A function $$f(x)$$ has an oblique asymptote $$g(x)=ax+b$$ when

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

Computation 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.