I know that the horizontal asymptotes of a function $f$ can be found by computing the limit of $f$ at $\pm \infty$, potentially using l'Hospital's rule when the limit takes the form $\infty/\infty$.
My question is, can l'Hospital's rule and limits be used to
- find a vertical asymptote
- verify that a vertical asymptope exists
A specific question I am looking at is $f(x) = \frac {x^2 -2}{(x-2)^2}$. I know that the vertical asymptote is 2. If (1) and (2) are possible could someone demonstrate the notation? Thank you very much.
HINT
Definition of $f$ having a vertical asymptote at some $x = a$ is that $$ \lim_{x \to a} f(x) = \pm \infty, $$ which implies that $f$ will be discontinuous there. Typically this happens in cases of fraction denominators turning $0$ or the function in some other way not existing at $x=a$...