Chapter 2.5, Problem 4E

For the function g whose graph is given, state the following.

(a) $\underset{x\to \infty }{\lim }g(x)$

(b) $\underset{x\to -\infty }{\lim }g(x)$

(c) $\underset{x\to 0}{\lim }g(x)$

(d) $\underset{x\to 2^{-}}{\lim }g(x)$

(e) $\underset{x\to 2^{+}}{\lim }g(x)$

(f) The equations of the asymptotes

To determine

To find: The value of $\underset{x\to \infty }{\lim }g\left(x\right)$.

Answer to Problem 4E

The value of $\underset{x\to \infty }{\lim }g\left(x\right)=2$.

Explanation of Solution

From the given graph, it is observed that the curve approaches $2$ as x approaches infinity from either side.

That is, $\underset{x\to \infty }{\lim }g\left(x\right)=2$.

To determine

To find: The value of $\underset{x\to -\infty }{\lim }g\left(x\right)$.

Answer to Problem 4E

Answers:

The value of $\underset{x\to -\infty }{\lim }g\left(x\right)=-1$.

Explanation of Solution

From the given graph, it is observed that the curve approaches $-1$ as x approaches negative infinity from either side.

Therefore, $\underset{x\to -\infty }{\lim }g\left(x\right)=-1$.

To determine

To find: The value of $\underset{x\to 0}{\lim }g\left(x\right)$.

Answer to Problem 4E

The value of $\underset{x\to 0}{\lim }g\left(x\right)=-\infty$.

Explanation of Solution

From the given graph, it is observed that the curve approaches $-\infty$ as x approaches 0 from either side.

Therefore, $\underset{x\to 0}{\lim }g\left(x\right)=-\infty$.

To determine

To find: The value of $\underset{x\to 2^{-}}{\lim }g\left(x\right)$.

Answer to Problem 4E

The value of $\underset{x\to 2^{-}}{\lim }g\left(x\right)=-\infty$.

Explanation of Solution

From the given graph, it is observed that the curve approaches $-\infty$ as x approaches 2 from the left side.

That is, $\underset{x\to 2^{-}}{\lim }g\left(x\right)=-\infty$.

To determine

To find: The value of $\underset{x\to 2^{+}}{\lim }g\left(x\right)$.

Answer to Problem 4E

The value of $\underset{x\to 2^{+}}{\lim }g\left(x\right)=\infty$.

Explanation of Solution

From the given graph, it is observed that the curve approaches $\infty$ as x approaches 2 from the right side.

That is, $\underset{x\to 2^{+}}{\lim }g\left(x\right)=\infty$.

To determine

To find: The equations of the asymptotes

Answer to Problem 4E

The equations of the asymptotes are $x=0,x=2,y=-1\text{ and }y=2$.

Explanation of Solution

Recall the definition that the line $x=a$ is said to be a vertical asymptote of the function $f\left(x\right)$, if either $\underset{x\to a^{-}}{\lim }f\left(x\right)=\pm \infty$ or $\underset{x\to a^{+}}{\lim }f\left(x\right)=\pm \infty$.

Recall the definition that the line $x=a$ is said to be a horizontal asymptote of the function $f\left(x\right)$, if either $\underset{x\to \pm \infty }{\lim }f\left(x\right)=-a$ or $\underset{x\to \pm }{\lim }f\left(x\right)=a$.

The limits obtained from part (a)-(e), it can be concluded that the equations of asymptote are $x=0,x=2,y=-1\text{ and }y=2$.