9. the graph of h in the figure has vertical asymptotes at x = -2 and x = 3. analyze the following…

Explanation:

Step1: Analyze left - hand limit as $x\to - 2$

As $x$ approaches $-2$ from the left side ($x\to - 2^{-}$), the function $h(x)$ decreases without bound. So, $\lim_{x\to - 2^{-}}h(x)=-\infty$.

Step2: Analyze right - hand limit as $x\to - 2$

As $x$ approaches $-2$ from the right side ($x\to - 2^{+}$), the function $h(x)$ increases without bound. So, $\lim_{x\to - 2^{+}}h(x)=\infty$.

Step3: Analyze two - sided limit as $x\to - 2$

Since $\lim_{x\to - 2^{-}}h(x)=-\infty$ and $\lim_{x\to - 2^{+}}h(x)=\infty$, the two - sided limit $\lim_{x\to - 2}h(x)$ does not exist.

Step4: Analyze left - hand limit as $x\to 3$

As $x$ approaches $3$ from the left side ($x\to 3^{-}$), the function $h(x)$ increases without bound. So, $\lim_{x\to 3^{-}}h(x)=\infty$.

Step5: Analyze right - hand limit as $x\to 3$

As $x$ approaches $3$ from the right side ($x\to 3^{+}$), the function $h(x)$ decreases without bound. So, $\lim_{x\to 3^{+}}h(x)=-\infty$.

Step6: Analyze two - sided limit as $x\to 3$

Since $\lim_{x\to 3^{-}}h(x)=\infty$ and $\lim_{x\to 3^{+}}h(x)=-\infty$, the two - sided limit $\lim_{x\to 3}h(x)$ does not exist.

Answer:

a. $\lim_{x\to - 2^{-}}h(x)=-\infty$ b. $\lim_{x\to - 2^{+}}h(x)=\infty$ c. $\lim_{x\to - 2}h(x)$ does not exist d. $\lim_{x\to 3^{-}}h(x)=\infty$ e. $\lim_{x\to 3^{+}}h(x)=-\infty$ f. $\lim_{x\to 3}h(x)$ does not exist