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) : Mathematics

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

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.

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.

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 :

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

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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