Chapter 4.6, Problem 88E

(a)

To determine

To find:Thevalue of the function $f\left(x\right)$ .

Answer to Problem 88E

The value of $f\left(x\right)$ for $x\to 0^{+}$ is $f\left(x\right)\to \infty$

Explanation of Solution

Given information:

The given function is $f\left(x\right)=\cot x$ .

Calculation:

The graph for the function $f\left(x\right)=\cot x$ from $-2\pi$ to $2\pi$ is shown in figure (1).

  

Figure (1)

For $x\to 0^{+}$

Using the above graph as $x$ approaches $0^{+}$ , $f\left(x\right)$ approaches positive infinity.

The value of $f\left(x\right)$ for $x\to 0^{+}$ is $f\left(x\right)\to \infty$

Therefore, the value of $f\left(x\right)$ for $x\to 0^{+}$ is $f\left(x\right)\to \infty$ .

(b)

To determine

To find:The value of the function $f\left(x\right)$ .

Answer to Problem 88E

The value of $f\left(x\right)$ for $x\to 0^{-}$ is $f\left(x\right)\to -\infty$

Explanation of Solution

Given information:

The given function is $f\left(x\right)=\cot x$ .

Calculation:

The graph for the function $f\left(x\right)=\cot x$ from $-2\pi$ to $2\pi$ is shown in figure (1).

  

Figure (1)

For $x\to 0^{-}$

Using the above graph as $x$ approaches $f\left(x\right)$ approaches negative infinity.

The value of $f\left(x\right)$ for $x\to 0^{-}$ is $f\left(x\right)\to -\infty$

Therefore, the value of $f\left(x\right)$ for $x\to 0^{-}$ is $f\left(x\right)\to -\infty$ .

(c)

To determine

To find:The value of the function $f\left(x\right)$ .

Answer to Problem 88E

The value of $f\left(x\right)$ for $x\to {\pi }^{+}$ is $f\left(x\right)\to \infty$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=\cot x$ .

Calculation:

The graph for the function $f\left(x\right)=\cot x$ from $-2\pi$ to $2\pi$ is shown in figure (1).

  

Figure (1)

For $x\to {\pi }^{+}$

Using the above graph as $x$ approaches ${\pi }^{+}$ , $f\left(x\right)$ approaches positive infinity.

The value of $f\left(x\right)$ for $x\to {\pi }^{+}$ is $f\left(x\right)\to \infty$

Therefore, the value of $f\left(x\right)$ for $x\to {\pi }^{+}$ is $f\left(x\right)\to \infty$ .

(d)

To determine

To find:The value of the function $f\left(x\right)$ .

Answer to Problem 88E

The value of $f\left(x\right)$ for $x\to {\pi }^{-}$ is $f\left(x\right)\to -\infty$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=\cot x$ .

Calculation:

The graph for the function $f\left(x\right)=\cot x$ from $-2\pi$ to $2\pi$ is shown in figure (1).

  

Figure (1)

For $x\to {\pi }^{-}$

Using the above graph as $x$ approaches ${\pi }^{-}$ , $f\left(x\right)$ approaches negative infinity.

The value of $f\left(x\right)$ for $x\to {\pi }^{-}$ is $f\left(x\right)\to -\infty$

Therefore, the value of $f\left(x\right)$ for $x\to {\pi }^{-}$ is $f\left(x\right)\to -\infty$ .