Answer
(a)$ \lim_{x\to \infty } f(x) = -2$
(b) $ \lim_{x\to- \infty } f(x) = 2$
(c)$ \lim_{x\to1 } f(x) = \infty$
(d) $ \lim_{x\to3 } f(x) =-\infty$
(e) $y=2$ , $y=-2$ , $x=1$ and $x=3$
Work Step by Step
(a) From the given figure , as $x \to \infty$ the graph of $f(x) $ seems to level out at $y = -2$, then $ \lim_{x\to \infty } f(x) = -2$
(b) From the given figure , as $x \to -\infty$ the graph of $f(x) $ seems to level out at $y =2$, then $ \lim_{x\to- \infty } f(x) = 2$
(c) From the given figure , as $x \to 1$ the graph of $f(x) $diverges towards $\infty$, then $ \lim_{x\to1 } f(x) =\infty$
(d) From the given figure , as $x \to 3$ the graph of $f(x) $diverges towards $-\infty$, then $ \lim_{x\to3 } f(x) =-\infty$
(e) From part (a) to (c) , the equations of asymptotes are $y=2$ , $y=-2$ , $x=1$ and $x=3$
