What are the asymptotes of f(x)=(1/(x-10))+(1/(x-20))?

Apr 10, 2018

y=0 if x =>+-oo, f(x) =-oo if x=>10^-, f(x)=+oo if x=>10^+, f(x) =-oo if x=>20^-, f(x)=+oo if x=>20^+

Explanation:

$f \left(x\right) = \frac{1}{x - 10} + \frac{1}{x - 20}$ let's find first limits.
Actually they are pretty obvious :
$L i m \left(x \to \pm \infty\right) f \left(x\right) = L i m \left(x \to \pm \infty\right) \frac{1}{x - 10} + \frac{1}{x - 20} = 0 + 0 = 0$ (when you divide a rational number by an infinite, the result is near to 0)
Now let's study limits in 10 and in 20.
$L i m \left(x \implies {10}^{-}\right) = \frac{1}{{0}^{-}} - \frac{1}{10} = - \infty$
$L i m \left(x \implies {20}^{-}\right) = \frac{1}{{0}^{-}} + \frac{1}{10} = - \infty$
$L i m \left(x \implies {10}^{+}\right) = \frac{1}{{0}^{+}} - \frac{1}{10} = + \infty$
$L i m \left(x \implies {20}^{-}\right) = \frac{1}{{0}^{+}} + \frac{1}{10} = + \infty$