Question

What are the asymptotes and removable discontinuities, if any, of `f(x)= (x^2+4)/(x-3)`?

Answer 1

No removable discontinuities, and the 2 asymptotes of this function are `x = 3` and `y = x`.

Explanation:

This function is not defined at `x = 3`, but you can still evaluate the limits on the left and on the right of `x = 3`.

`lim_(x->3^-)f(x) = -oo` because the denominator will be strictly negative, and `lim_(x->3^+)f(x) = +oo` because the denomiator will be strictly positive, making `x = 3` an asymptote of `f`.

For the 2nd one, you need to evaluate `f` near the infinities. There is a property of rational functions telling you that only the biggest powers matter at the infinities, so it means that `f` will be equivalent to `x^2/x = x` at the infinites, making `y = x` another asymptote of `f`.

You can't remove this discontinuity, the 2 limits at `x=3` are different.

Here is a graph :
graph{(x^2 + 4)/(x - 3) [-163.5, 174.4, -72.7, 96.2]}