Question

How do you find the vertical, horizontal and slant asymptotes of: `y=1/(2-x)`?

Answer 1

vertical asymptote at x = 2
horizontal asymptote at y = 0

Explanation:

The denominator of y cannot be zero as this would make y 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: `2-x=0rArrx=2" is the asymptote"`

Horizontal asymptotes occur as

`lim_(xto+-oo),ytoc" (a constant)"`

divide terms on numerator/denominator by x

`y=(1/x)/(2/x-x/x)=(1/x)/(2/x-1)`

as `xto+-oo,yto0/(0-1)`

`rArry=0" is the asymptote"`

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here ( numerator-degree 0, denominator-degree 1 ) Hence there are no slant asymptotes.
graph{1/(2-x) [-10, 10, -5, 5]}