How do you find the vertical, horizontal or slant asymptotes for f(x)=(2x-3)/(x^2+2)?

Jul 13, 2017

$\text{horizontal asymptote at } y = 0$

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.

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

$\text{this has no real solutions hence there are no }$
$\text{vertical asymptotes}$

$\text{horizontal asymptotes occur as }$

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

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

$f \left(x\right) = \frac{\frac{2 x}{x} ^ 2 - \frac{3}{x} ^ 2}{{x}^{2} / {x}^{2} + \frac{2}{x} ^ 2} = \frac{\frac{2}{x} - \frac{3}{x} ^ 2}{1 + \frac{2}{x} ^ 2}$

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

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

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes.
graph{(2x-3)/(x^2+2) [-10, 10, -5, 5]}