Question

How do you find the vertical, horizontal and slant asymptotes of: `(x-3)/(x-2)`?

Answer 1

Use the fact that `(x-3)/(x-2) = (x-2-1)/(x-2) = (x-2)/(x-2) - 1/(x-2) = 1 - 1/(x-2)`.

Explanation:

I suppose you meant oblique asymptote, which intuitively refers to the 'slant' asymptote.

To find asymptotes, we check the `x` and `y` values where the function is undefined.

When `x-2=0`, the function is undefined. Thus the vertical asymptote is `x=2`.

Also, notice that as `x` gets larger, `1/(x-2)` gets smaller, but never zero. Thus, the function will never take on the value `y=1`, and thus that is the vertical asymptote.