Chapter 12.1, Problem 73E

To determine

To graph: the tangent function and to find $\underset{x\to 0}{\lim }\tan x$ and $\underset{x\to \frac{\pi }{4}}{\lim }\tan x$ . The conclusion about the existence of $\underset{x\to \frac{\pi }{2}}{\lim }\tan x$ .

Answer to Problem 73E

When $x=0,f\left(x\right)=0$

When $x=\frac{\pi }{4},f\left(x\right)=1$

When $x=\frac{\pi }{2},f\left(x\right)$ is not defined.

Explanation of Solution

Given information:

Calculation:

Consider the following function:

  $f\left(x\right)=\tan x$

To graph the function using graphing utility TI-83, proceed as follows.

Press $Y=$ and key in the function. The display will be

  

Press windows and choose the proper scale. The display will be

  

Now press graph. The display will be

  

To find the value of the function at a defined point, proceed as follows.

Press $2^{nd}$ trace. The display will be:

  

Now, provide the required value, to get the value of the function at the point.

When $x=0$

  

When $x=\frac{\pi }{4}$

  

When $x=\frac{\pi }{2}$

  

Hence

When $x=0,f\left(x\right)=0$

When $x=\frac{\pi }{4},f\left(x\right)=1$

When $x=\frac{\pi }{2},f\left(x\right)$ is not defined.