How do you find the asymptotes for f(x)=(-7x + 5) / (x^2 + 8x -20)?

Jul 28, 2016

vertical asymptotes x = -10 , x = 2
horizontal asymptote y = 0

Explanation:

The denominator of f(x) cannot be zero as this would be 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 of x then they are vertical asymptotes.

solve: ${x}^{2} + 8 x - 20 = 0 \Rightarrow \left(x + 10\right) \left(x - 2\right) = 0$

$\Rightarrow x = - 10 , x = 2 \text{ are the asymptotes}$

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 ${x}^{2}$

$\frac{\frac{- 7 x}{x} ^ 2 + \frac{5}{x} ^ 2}{{x}^{2} / {x}^{2} + \frac{8 x}{x} ^ 2 - \frac{20}{x} ^ 2} = \frac{- \frac{7}{x} + \frac{5}{x} ^ 2}{1 + \frac{8}{x} - \frac{20}{x} ^ 2}$

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

$\Rightarrow y = 0 \text{ is the asymptote}$
graph{(7x+5)/(x^2+8x-20) [-40, 40, -20, 20]}