# How do you find the asymptotes for R(x)= (x+2)/(x^2-64)?

Dec 28, 2016

See explanation

#### Explanation:

The expression becomes undefined at the point where you have:

$\frac{\text{some value}}{0}$.

So we have:

$\textcolor{b l u e}{{\lim}_{{x}^{2} - 64 \to {0}^{+}} = {\lim}_{x \to {8}^{+}} \frac{x + 2}{{x}^{2} - 64} \to + \infty}$

$\textcolor{b l u e}{{\lim}_{{x}^{2} - 64 \to {0}^{-}} = {\lim}_{x \to {8}^{-}} \frac{x + 2}{{x}^{2} - 64} \to - \infty}$

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Now we investigate as $x \to \pm \infty$

As $x$ becomes increasingly positive or negative the less and less influence is applied by the 2 in the numerator and the -64 in the denominator.

Thus the expression tend towards $\frac{x}{x} ^ 2 = \frac{1}{\pm x}$

And as $| x | \to \infty$ then $\frac{1}{| x |} \to 0$

$\textcolor{b l u e}{{\lim}_{x \to {\infty}^{-}} \frac{x + 2}{{x}^{2} - 64} \to {0}^{-}}$

$\textcolor{b l u e}{{\lim}_{x \to {\infty}^{+}} \frac{x + 2}{{x}^{2} - 64} \to {0}^{+}}$ 