Question

How do you find the vertical, horizontal and slant asymptotes of: `h(x)= (x^2-4)/(x)`?

Answer 1

vertical asymptote at x = 0
slant asymptote y = x

Explanation:

The denominator of h(x) cannot be zero as this would make h(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=0rArrx=0" is the asymptote"`

`color(blue)"Horizontal asymptotes"` occur when the degree of the numerator ≤ degree of the denominator. This is not the case here (numerator degree 2 , denominator degree 1 ) Hence there are no horizontal asymptotes.

`color(blue)"slant asymptotes"` occur when the degree of the numerator > degree of denominator. Hence there is a slant asymptote.

divide numerator by denominator.

`rArrh(x)=(x^2)/x-4/x=x-4/x`

as `xto+-oo,h(x)tox-0`

`rArry=x" is the asymptote"`
graph{(x^2-4)/x [-10, 10, -5, 5]}