# How do you find the vertical, horizontal and slant asymptotes of: (x^2-1)/(x^2+4)?

Apr 28, 2016

vertical asymptote: does not exist
horizontal asymptote: $f \left(x\right) = 1$
slant asymptote: does not exist

#### Explanation:

Finding the Vertical Asymptote

Given,

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

Factor the numerator.

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

Cancel out any factors that appear in the numerator and denominator. Since there aren't any, set the denominator equal to $0$ and solve for $x$.

${x}^{2} + 4 = 0$

${x}^{2} = - 4$

$x = \pm \sqrt{- 4}$

Since you can't take the square root of a negative number in the domain of real numbers, there is no vertical asymptote.

$\therefore$, the vertical asymptote does not exist.

Finding the Horizontal Asymptote

Given,

$f \left(x\right) = \frac{\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{1} {x}^{2} - 1}{\textcolor{p u r p \le}{1} {x}^{2} + 4}$

Divide the $\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{\text{leading coefficient}}$ of the leading term in the numerator by the $\textcolor{p u r p \le}{\text{leading coefficient}}$ of the leading term in the denominator.

$f \left(x\right) = \frac{\textcolor{\mathrm{da} r k \mathmr{and} a n \ge}{1}}{\textcolor{p u r p \le}{1}}$

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{f \left(x\right) = 1} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

Finding the Slant Asymptote

Given,

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

There would be a slant asymptote if the degree of the leading term in the numerator was $1$ value larger than the degree of the leading term in the denominator. In your case, we see that the degree in both the numerator and denominator are equal.

$\therefore$, the slant asymptote does not exist.