Chapter 4.6, Problem 91E

a.

To determine

Use the graph to determine the behaviour of the function as $\text{x}\to \text{c}.$

Answer to Problem 91E

The function goes to infinity as $\text{x}$ approaches $\text{0}$ from the right.

Explanation of Solution

Given information:

Use a graphing utility to graph the function. Use the graph to determine the behaviour of the function as $\text{x}\to \text{c}.$ (Note: The notation $\text{x}\to {\text{c}}^{+}$ indicates that $\text{x}$ approaches $c$ from the right and $\text{x}\to {\text{c}}^{-}$ indicates that $\text{x}$ approaches $c$ from the left.)

  $\text{x}\to 0^{+}$

Calculation:

Use the graphing utility to graph the function and determine the behaviour of the function $\text{x}\to 0^{+},x\to 0^{-},x\to {\pi }^{+}$ and $x\to {\pi }^{-}$

  $f\left(x\right)=\cot x$

The function is $f\left(x\right)=\cot x$ . Use the graphing utility to graph the function

For $\text{x}\to 0^{+}$ :

With the help of the above graph the function goes to infinity as $\text{x}$ approaches $\text{0}$ from the right.

Hence, the function goes to infinity as $\text{x}$ approaches $\text{0}$ from the right.

b.

To determine

Use the graph to determine the behaviour of the function as $\text{x}\to \text{c}.$

Answer to Problem 91E

The function goes to negative infinity as $\text{x}$ approaches $\text{0}$ from the left.

Explanation of Solution

Given information:

Use a graphing utility to graph the function. Use the graph to determine the behaviour of the function as $\text{x}\to \text{c}.$ (Note: The notation $\text{x}\to {\text{c}}^{+}$ indicates that $\text{x}$ approaches $c$ from the right and $\text{x}\to {\text{c}}^{-}$ indicates that $\text{x}$ approaches $c$ from the left.)

  $x\to 0^{-}$

Calculation:

Use the graphing utility to graph the function and determine the behaviour of the function $\text{x}\to 0^{+},x\to 0^{-},x\to {\pi }^{+}$ and $x\to {\pi }^{-}$

The function is $f\left(x\right)=\cot x$ . Use the graphing utility to graph the function.

   For $x\to 0^{-}$ :

With the help of the above graph the function goes to negative infinity as $\text{x}$ approaches $\text{0}$ from the left.

Hence, the function goes to negative infinity as $\text{x}$ approaches $\text{0}$ from the left.

c.

To determine

Use the graph to determine the behaviour of the function as $\text{x}\to \text{c}.$

Answer to Problem 91E

The function goes to infinity as $\text{x}$ approaches $\pi$ from the right.

Explanation of Solution

Given information:

Use a graphing utility to graph the function. Use the graph to determine the behaviour of the function as $\text{x}\to \text{c}.$ (Note: The notation $\text{x}\to {\text{c}}^{+}$ indicates that $\text{x}$ approaches $c$ from the right and $\text{x}\to {\text{c}}^{-}$ indicates that $\text{x}$ approaches $c$ from the left.)

  $x\to {\pi }^{+}$

Calculation:

Use the graphing utility to graph the function and determine the behaviour of the function $\text{x}\to 0^{+},x\to 0^{-},x\to {\pi }^{+}$ and $x\to {\pi }^{-}$

The function is $f\left(x\right)=\cot x$ . Use the graphing utility to graph the function.

   For $x\to {\pi }^{+}$

With the help of the above graph the function goes to infinity as $\text{x}$ approaches $\pi$ from the right.

Hence, the function goes to infinity as $\text{x}$ approaches $\pi$ from the right.

d.

To determine

Use the graph to determine the behaviour of the function as $\text{x}\to \text{c}.$

Answer to Problem 91E

The function goes to infinity as $\text{x}$ approaches $\pi$ from the left.

Explanation of Solution

Given information:

Use a graphing utility to graph the function. Use the graph to determine the behaviour of the function as $\text{x}\to \text{c}.$ (Note: The notation $\text{x}\to {\text{c}}^{+}$ indicates that $\text{x}$ approaches $c$ from the right and $\text{x}\to {\text{c}}^{-}$ indicates that $\text{x}$ approaches $c$ from the left.)

  $x\to {\pi }^{-}$

Calculation:

Use the graphing utility to graph the function and determine the behaviour of the function $\text{x}\to 0^{+},x\to 0^{-},x\to {\pi }^{+}$ and $x\to {\pi }^{-}$

The function is $f\left(x\right)=\cot x$ . Use the graphing utility to graph the function.

  

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

With the help of the above graph the function goes to negative infinity as $\text{x}$ approaches $\pi$ from the left.

Hence, the function goes to infinity as $\text{x}$ approaches $\pi$ from the left.