Chapter 2.5, Problem 3E

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

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

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

(c) $\underset{x\to 1}{\lim }f(x)$

(d) $\underset{x\to 3}{\lim }f(x)$

(e) The equations of the asymptotes

To determine

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

Answer to Problem 3E

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

Explanation of Solution

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

To determine

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

Answer to Problem 3E

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

Explanation of Solution

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

To determine

To find: The value of $\underset{x\to 1}{\lim }f\left(x\right)$.

Answer to Problem 3E

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

Explanation of Solution

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

To determine

To find: The value of $\underset{x\to 3}{\lim }f\left(x\right)$.

Answer to Problem 3E

The value of $\underset{x\to 3}{\lim }f\left(x\right)=-\infty$.

Explanation of Solution

From the given graph, it is observed that the curve approaches $-\infty$ as x approaches 3 from either side.

To determine

To find: The equations of the asymptotes.

Answer to Problem 3E

Solution: The equations of the asymptote are $x=1,x=3,y=2\text{ and }y=-2$.

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 $\underset{x\to \pm }{\lim }f\left(x\right)=a$.

From the limits obtained from the parts (a)-(d), it can be concluded that the equations of asymptote are $x=1,x=3,y=2\text{ and }y=-2$.