# How do you find the vertical, horizontal or slant asymptotes for f(x) = (4x) / (x^2+1)?

Sep 19, 2016

horizontal asymptote at y = 0

#### Explanation:

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

solve : ${x}^{2} + 1 = 0 \Rightarrow {x}^{2} = - 1$

This has no real solutions hence there are no vertical asymptotes.

Horizontal asymptotes occur as

${\lim}_{x \to \pm \infty} , f \left(x\right) \to c \text{ (a constant)}$

divide all terms on numerator/denominator by the highest power of x that is ${x}^{2}$

$f \left(x\right) = \frac{\frac{4 x}{x} ^ 2}{{x}^{2} / {x}^{2} + \frac{1}{x} ^ 2} = \frac{\frac{4}{x}}{1 + \frac{1}{x} ^ 2}$

as $x \to \pm \infty , f \left(x\right) \to \frac{0}{1 + 0}$

$\Rightarrow y = 0 \text{ is the asymptote}$

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here ( numerator-degree 1 , denominator-degree 2 ) Hence there are no slant asymptotes.
graph{(4x)/(x^2+1) [-10, 10, -5, 5]}