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.

Asymptote of a Function - dCode

Tag(s) : Functions

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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.

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

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

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**.

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**.

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**.

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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