Asymptote of a Function

Question 1

Answer

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

Question 2

Answer

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.

Question 3

Answer

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.

Question 4

Answer

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.

Question 5

Answer

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.

Asymptotes Calculator