Chapter 4.6, Problem 61E

a.

To determine

To graph: The given function using a graphing utility and then to find the value of $f(x)$ as $x\to 0^{+}$ .

Answer to Problem 61E

As $x\to 0^{+},f(x)\to \infty$ .

Explanation of Solution

Given Information: Given function is $f(x)=\csc x$ .

Graph:

  

Interpretation:

From the graph, as $x\to 0^{+},f(x)\to \infty$ .

b.

To determine

To graph: The given function using a graphing utility and then to find the value of $f(x)$ as $x\to 0^{-}$ .

Answer to Problem 61E

As $x\to 0^{-},f(x)\to -\infty$ .

  $\underset{x\to 0}{\lim }(\csc x)$ does not exist.

Explanation of Solution

Given Information: Given function is $f(x)=\csc x$ .

Graph:

  

Interpretation:

From the graph, as $x\to 0^{-},f(x)\to -\infty$ .

Since the left sided and right sided limits are not equal, so $\underset{x\to 0}{\lim }(\csc x)$ does not exist.

c.

To determine

To graph: The given function using a graphing utility and then to find the value of $f(x)$ as $x\to {\pi }^{+}$ .

Answer to Problem 61E

As $x\to {\pi }^{+},f(x)\to -\infty$ .

Explanation of Solution

Given Information: Given function is $f(x)=\csc x$ .

Graph:

  

Interpretation:

From the graph, as $x\to {\pi }^{+},f(x)\to -\infty$ .

d.

To determine

To graph: The given function using a graphing utility and then to find the value of $f(x)$ as $x\to {\pi }^{-}$ .

Answer to Problem 61E

As $x\to {\pi }^{-},f(x)\to \infty$ .

  $\underset{x\to \pi }{\lim }(\csc x)$ does not exist.

Explanation of Solution

Given Information: Given function is $f(x)=\csc x$ .

Graph:

  

Interpretation:

From the graph, as $x\to {\pi }^{-},f(x)\to \infty$ .

Since the left sided and right sided limits are not equal, so $\underset{x\to \pi }{\lim }(\csc x)$ does not exist.