Precalculus
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# How do you find vertical, horizontal and oblique asymptotes for (x^2-9x+4)/(x+6)?

May 31, 2016

The vertical asymptote as at $\text{ } x = - 6$
The oblique asymptote is $\text{ } y = x - 15$

#### Explanation:

Given:$\text{ } \frac{{x}^{2} - 9 x + 4}{x + 6}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the vertical asymptote}}$

The equation is undefined at the denominator being 0

So the excluded value is that which gives $1 + 6 = 0$

$\textcolor{g r e e n}{\text{So the vertical asymptote as at } x = - 6}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Determine the oblique asymptote}}$

Investigate the extreme values of $x$

$\textcolor{b r o w n}{\text{I can not spot how to simplify the given expression so by}}$
$\textcolor{b r o w n}{\text{polynomial division we have } y = x - 15 + \frac{94}{x + 6}}$

$y = {\lim}_{x \to {\textcolor{w h i t e}{}}^{-} \infty} x - 15 + \frac{94}{x + 6} \text{ "=" } x - 15$

$\textcolor{g r e e n}{\text{So the oblique asymptote is } y = x - 15}$

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