Question

1. the graph of p in the figure has vertical asymptotes at x = -2 and x = 3. analyze the following limits.

a. $lim_{x
ightarrow - 2^{-}}p(x)$ b. $lim_{x
ightarrow - 2^{+}}p(x)$ c. $lim_{x
ightarrow - 2}p(x)$
d. $lim_{x
ightarrow 3^{-}}p(x)$ e. $lim_{x
ightarrow 3^{+}}p(x)$ f. $lim_{x
ightarrow 3}p(x)$

Explanation:

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

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

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

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

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

Since $\lim_{x\to - 2^{-}}p(x)=-\infty$ and $\lim_{x\to - 2^{+}}p(x)=\infty$, the two - sided limit $\lim_{x\to - 2}p(x)$ 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 $p(x)$ decreases without bound. So, $\lim_{x\to 3^{-}}p(x)=-\infty$.

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

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

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

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

Answer:

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