Question

How do you find the asymptotes for `y=(x^3 +3x^2-4x-10)/(x^2-4)`?

Answer 1

Consider behaviour in the infinite domain limits and when the denominator goes to zero. The function has three asymptotic lines, `y=x`, `x=-2`, and `x=+2`.

Explanation:

In the limit as `xrarroo`, `yrarr x^3/x^2=x`.
In the limit as `xrarr-oo`, also `yrarr x^3/x^2=x`. So in both infinite domain limits the function tends to the diagonal straight line `y=x`, an asymptote in both cases.

The function has two poles on the real line, at `x=+-2`, where the denominator is zero and the numerator is non-zero. So the two lines `x=+-2` are vertical asymptotes of the functions.