Chapter 4.6, Problem 93E

(a)

To determine

The behaviour of the function using graphical utility.

Answer to Problem 93E

The value is $x\to \frac{{\pi }^{+}}{2},f\left(x\right)\to -\infty$

Explanation of Solution

Given information:

The graph to determine the behaviour of the function as $x\to c$

  $x\to {\left(\frac{\pi }{2}\right)}^{+}$

Formula used:

Maple software is used.

Calculation:

Let us plot the graph for $f\left(x\right)=\tan x\text{from }-2\pi \text{to}2\pi$ using the command

  $plot\left(\tan \left(x\right),x=-2\pi \mathrm{...2}\pi \right)$

In the software Maple

  

The above graph represents the equation $f\left(x\right)=\tan x$

Also, from the above graph we will determine the behaviour of the function as $x\to c$

  $x\to {\left(\frac{\pi }{2}\right)}^{+}$

From the above graph we can see that as x approaches $\frac{{\pi }^{+}}{2},f\left(x\right)$ approaches negative infinity.

Therefore, $x\to \frac{{\pi }^{+}}{2},f\left(x\right)\to -\infty$

Conclusion:

The value is $x\to \frac{{\pi }^{+}}{2},f\left(x\right)\to -\infty$

(b)

To determine

The behaviour of the function using graphical utility.

Answer to Problem 93E

The value is $x\to \frac{{\pi }^{-}}{2},f\left(x\right)\to \infty$

Explanation of Solution

Given information:

The graph to determine the behaviour of the function as $x\to c$

  $x\to {\left(\frac{\pi }{2}\right)}^{-}$

Formula used:

Maple software is used

Calculation:

Let us plot the graph for $f\left(x\right)=\tan x\text{from }-2\pi \text{to}2\pi$ using the command

  $plot\left(\tan \left(x\right),x=-2\pi \mathrm{...2}\pi \right)$

In the software Maple

  

The above graph represents the equation $f\left(x\right)=\tan x$

Also, from the above graph we will determine the behaviour of the function as $x\to c$

  $x\to {\left(\frac{\pi }{2}\right)}^{-}$

From the above graph we can see that as x approaches $\frac{{\pi }^{-}}{2}$ , $f\left(x\right)$ approaches positive infinity.

Therefore, $x\to \frac{{\pi }^{-}}{2},f\left(x\right)\to \infty$

Conclusion:

The value is $x\to \frac{{\pi }^{-}}{2},f\left(x\right)\to \infty$

(c)

To determine

The behaviour of the function using graphical utility.

Answer to Problem 93E

The value is $x\to -\frac{{\pi }^{+}}{2},f\left(x\right)\to -\infty$

Explanation of Solution

Given information:

The graph to determine the behaviour of the function as $x\to c$

  $x\to {\left(-\frac{\pi }{2}\right)}^{+}$

Formula used:

Maple software is used.

Calculation:

Let us plot the graph for $f\left(x\right)=\tan x\text{from }-2\pi \text{to}2\pi$ using the command

  $plot\left(\tan \left(x\right),x=-2\pi \mathrm{...2}\pi \right)$

In the software Maple

  

The above graph represents the equation $f\left(x\right)=\tan x$

Also, from the above graph we will determine the behaviour of the function as $x\to c$

  $x\to {\left(-\frac{\pi }{2}\right)}^{+}$

From the above graph we can see that as x approaches $-\frac{{\pi }^{+}}{2}$ , $f\left(x\right)$ approaches negative infinity.

Therefore, $x\to -\frac{{\pi }^{+}}{2},f\left(x\right)\to -\infty$

Conclusion:

The value is $x\to -\frac{{\pi }^{+}}{2},f\left(x\right)\to -\infty$

(d)

To determine

The behaviour of the function using graphical utility.

Answer to Problem 93E

The value is $x\to -\frac{{\pi }^{-}}{2},f\left(x\right)\to \infty$

Explanation of Solution

Given information:

The graph to determine the behaviour of the function as $x\to c$

  $x\to {\left(-\frac{\pi }{2}\right)}^{-}$

Formula used:

Maple software is used

Calculation:

Let us plot the graph for $f\left(x\right)=\tan x\text{from }-2\pi \text{to}2\pi$ using the command

  $plot\left(\tan \left(x\right),x=-2\pi \mathrm{...2}\pi \right)$

In the software Maple

  

The above graph represents the equation $f\left(x\right)=\tan x$

Also, from the above graph we will determine the behaviour of the function as $x\to c$

  $x\to {\left(-\frac{\pi }{2}\right)}^{-}$

From the above graph we can see that as x approaches $-\frac{{\pi }^{-}}{2}$ , $f\left(x\right)$ approaches positive infinity.

Therefore, $x\to -\frac{{\pi }^{-}}{2},f\left(x\right)\to \infty$

Conclusion:

The value is $x\to -\frac{{\pi }^{-}}{2},f\left(x\right)\to \infty$