Textbook Problem

Each of Problems 21-24 contains a function and its graph. For each problem, answer parts (a) and (b).

(a) Use the graph to determine, as well as you can,

(i) vertical asymptotes, (ii) lim $\underset{x\to \infty }{\lim }f(x)$.

(iii) $\underset{x\to -\infty }{\lim }f(x)$

. (iv) horizontal asymptotes.

(b) Check your conclusions in (a) by using the functions to determine items (i)-(iv) analytically

$f(x)=\frac{2{(x+1)}^3(x+5)}{{(x-3)}^2{(x+2)}^2}$