# How do you find the vertical, horizontal and slant asymptotes of:  f(x)=(1/(x-10))+(1/(x-20))?

May 3, 2018

Vertical asymptotes: $x = 20$.

Horizontal asymptotes: $y = 0$

Slant asymptotes: none.

#### Explanation:

Vertical asymptotes: you have to look for the points in which the function is not defined. The way the function is written, it is really easy to see that the first denominator vanished for $x = 10$, and the second for $x = 20$.

Horizontal asymptotes: you have to consider the limit as $x$ approaches $\setminus \pm \setminus \infty$. Since in both cases you have a behaviour like $\frac{1}{\setminus \pm \setminus \infty}$, the horizontal aymptote is the $x$ axis, i.e. $y = 0$

Slant asymptotes: since you have horizontal asymptotes, you can't have slant asymptotes.