Question

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

Answer 1

`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(x) =1/(x-10)+1/(x-20)` let's find first limits.
Actually they are pretty obvious :
`Lim(x->+-oo) f(x)=Lim(x->+-oo) 1/(x-10)+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.
`Lim(x=>10^-)=1/(0^-)-1/10=-oo`
`Lim(x=>20^-)=1/(0^-)+1/10=-oo`
`Lim(x=>10^+)=1/(0^+)-1/10=+oo`
`Lim(x=>20^-)=1/(0^+)+1/10=+oo`
\0/ here's our answer !