Question

How do you find the vertical, horizontal and slant asymptotes of: `f(x)=x/(x-7)`?

Answer 1

vertical asymptote at x = 7
horizontal asymptote at y = 1

Explanation:

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: `x-7=0rArrx=7" is the asymptote"`

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

`f(x)=(x/x)/(x/x-7/x)=1/(1-7/x)`

as `xto+-oo,f(x)to1/(1-0)`

`rArry=1" is the asymptote"`

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here ( both degree 1 ) Hence there are no slant asymptotes.
graph{x/(x-7) [-20, 20, -10, 10]}