Answer
a) $\lim\limits_{x \to -2}f(x) $ $= \infty$.
b) $\lim\limits_{x \to 0^{-}}f(x) = \infty$
c) $\lim\limits_{x \to 0^{+}}f(x) = 2$
d) $\lim\limits_{x \to 2^{-}}f(x) = 2$
e) $\lim\limits_{x \to 2^{+}}f(x) = -\infty$
f) $x = -2$, $x=0$ and $x=2$
Work Step by Step
a) On either side of $x=-2$, the function grows upwards asymptotically. We assume this trend continues up forever and the lines never meet. Our left-hand side and right-hand side limits will be equal to infinity, so the overall limit is infinity.
b) We observe the same asymptotic growth as the function approaches 0 from the left-hand-side; therefore, the half-limit is equal to infinity.
c) Approaching 0 from the right-hand-side, we have oscillations decreasing in amplitude. We assume that this function does converge to one point as it approaches $x=0$ and gets a right hand limit of 2
d) We follow the function towards 2 from the left hand side (as its oscillations increase in amplitude) and find that it trends towards 2 at $x=2$
e) From the right hand side of $x=2$, the function shows asymptotic decrease, so the half-limit's value will be negative infinity.
f) We find our vertical asymptotes at the x-values where at least one half-limit's value is infinite (that is, where we see asymptotic growth towards $\pm \infty$).
