Answer
(a)$ \lim_{x\to \infty } g(x) = 2$
(b) $ \lim_{x\to- \infty } g(x) = -1$
(c) $ \lim_{x\to0} g(x) =-\infty$
(d) $ \lim_{x\to2^- } g(x) =-\infty$
(e)$ \lim_{x\to2^+ } g(x) = \infty$
(f) $y=2$ , $y=-1$ , $x=0$ and $x=2$
Work Step by Step
(a) From the given figure , as $x \to \infty$ the graph of $g(x) $ seems to level out at $y = 2$, then $ \lim_{x\to \infty } g(x) = 2$
(b) From the given figure , as $x \to -\infty$ the graph of $g(x) $ seems to level out at $y =-1$, then $ \lim_{x\to- \infty } g(x) = -1$
(c) From the given figure , as $x \to 0$ the graph of $g(x) $diverges towards $-\infty$, then $ \lim_{x\to0} g(x) =-\infty$
(d) From the given figure , as $x \to2^-$ the graph of $g(x) $diverges towards $-\infty$, then $ \lim_{x\to2^- } g(x) =-\infty$
(e)From the given figure , as $x \to2^+$ the graph of $g(x) $diverges towards $ \infty$, then $ \lim_{x\to2^+ } g(x) = \infty$
(f) From part (a) to (e) , the equations of asymptotes are $y=2$ , $y=-1$ , $x=0$ and $x=2$
