Question

What are the asymptotes and removable discontinuities, if any, of `f(x)= (4x)/(22-40x)`?

Answer 1

vertical asymptote `x=11/20`
horizontal asymptote y`=-1/10`

Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.

solve : `22-40x=0rArr40x=22rArrx=22/40=11/20`

`rArrx=11/20" is the asymptote"`

Horizontal asymptotes occur as

`lim_(xto+-oo),f(x)toc" (a constant)"`

divide terms on numerator/denominator by x

`((4x)/x)/(22/x-(40x)/x)=4/(22/x-40)`

as `xto+-oo,f(x)to4/(0-40)`

`rArry=4/(-40)=-1/10" is the asymptote"`

There are no removable discontinuities
graph{(4x)/(22-40x) [-10, 10, -5, 5]}