Answer
a) The limit does not exist.
b) $x = -2$
c) From parts a) and b), we can conclude that the vertical asymptote exists where the function is not defined (Here, where the denominator becomes $0$ and the function tends to $\pm\infty$)
Work Step by Step
a) For $f(x) = \frac{3x}{(x+2)^3}$ we have:
$\lim\limits_{x \to -2^-}\frac{3x}{(x+2)^3} = \infty$
whereas:
$\lim\limits_{x \to -2^+}\frac{3x}{(x+2)^3} = -\infty$
Since left-hand limit and right-hand limit are not equal, the limit does not exist.
b) In our case there is a vertical asymptote:
$x=-2$
c) The vertical asymptote is where for a given $x$-value, a $y$-value does not exist. It is where, in the graph, one can draw a straight, vertical line which does not touch the function (refer to image).