Question

1. the graph of g in the figure has vertical asymptotes at x = 2 and x = 4. analyze the following limits. a. $lim_{x

ightarrow2^{-}}g(x)$ b. $lim_{x
ightarrow2^{+}}g(x)$ c. $lim_{x
ightarrow2}g(x)$ d. $lim_{x
ightarrow4^{-}}g(x)$ e. $lim_{x
ightarrow4^{+}}g(x)$ f. $lim_{x
ightarrow4}g(x)$

Explanation:

Step1: Analyze left - hand limit as $x\to2$

As $x$ approaches $2$ from the left side ($x\to2^{-}$), the graph of $y = g(x)$ goes to positive infinity. So, $\lim_{x\to2^{-}}g(x)=\infty$.

Step2: Analyze right - hand limit as $x\to2$

As $x$ approaches $2$ from the right side ($x\to2^{+}$), the graph of $y = g(x)$ goes to positive infinity. So, $\lim_{x\to2^{+}}g(x)=\infty$.

Step3: Analyze two - sided limit as $x\to2$

Since $\lim_{x\to2^{-}}g(x)=\lim_{x\to2^{+}}g(x)=\infty$, then $\lim_{x\to2}g(x)=\infty$.

Step4: Analyze left - hand limit as $x\to4$

As $x$ approaches $4$ from the left side ($x\to4^{-}$), the graph of $y = g(x)$ goes to negative infinity. So, $\lim_{x\to4^{-}}g(x)=-\infty$.

Step5: Analyze right - hand limit as $x\to4$

As $x$ approaches $4$ from the right side ($x\to4^{+}$), the graph of $y = g(x)$ goes to positive infinity. So, $\lim_{x\to4^{+}}g(x)=\infty$.

Step6: Analyze two - sided limit as $x\to4$

Since $\lim_{x\to4^{-}}g(x)
eq\lim_{x\to4^{+}}g(x)$, the two - sided limit $\lim_{x\to4}g(x)$ does not exist.

Answer:

a. $\lim_{x\to2^{-}}g(x)=\infty$
b. $\lim_{x\to2^{+}}g(x)=\infty$
c. $\lim_{x\to2}g(x)=\infty$
d. $\lim_{x\to4^{-}}g(x)=-\infty$
e. $\lim_{x\to4^{+}}g(x)=\infty$
f. $\lim_{x\to4}g(x)$ does not exist