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Asymptote of a Function

Tool to find the equations of the asymptotes (horizontal, vertical, oblique or curved) of a function or mathematical expression.

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Asymptote of a Function -

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# Asymptote of a Function

## Answers to Questions (FAQ)

### What is an asymptote? (Definition)

An asymptote is a line (or sometimes a curve) which tends (similarly to a tangent) to the function at infinity.

### How to find a horizontal asymptote?

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

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

To find a horizontal asymptote, the calculation of this limit is a sufficient condition.

Example: $1/x$ has for asymptote $y=0$ 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).

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

To find a horizontal asymptote, the calculation of this limit is a sufficient condition.

Example: $1/x$ has for asymptote $x=0$ 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.

For a rational function (with a fraction: numerator over denominator), values for which the denominator is zero are 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:

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

For a rational function applying a polynomial division allows to find an oblique asymptote.

### 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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