Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `f(x)= (x-4)/(x^2-8x+16)`?

Answer 1

`x=4` is a Vertical Asymptote

`y=0` is a Horizontal Asymptote

There is no Slant Asymptote.

Explanation:

This function is defined for all `x in RR` except `x=4`

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

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

Domain of `f` is:`(-oo,4)U(4,+oo)`

The vertical Asymptote is computed by setting the denominator to zero

`x-4=0rArrcolor(blue)(x=4)` is a `color(blue)( V.A )`

Let us start computing the asymptote:

The degree of numerator is greater than that of the denominator

Therefore ,`color(blue)(y=0)` is a `color(blue)(H.A)`

There is no Slant asymptote.