Chapter 2.1, Problem 28E

To determine

To find: The value of $\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)$ with the help of graph and check the results algebraically.

Answer to Problem 28E

The value of $\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)$ is $-0.25$ .

Explanation of Solution

Given information: The limit is $\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)$ .

Calculation:

Let $y=\frac{\frac{1}{2+x}-\frac{1}{2}}{x}$ to find the value of $\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)$ graphically.

To graph a function $y=\frac{\frac{1}{2+x}-\frac{1}{2}}{x}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\frac{\frac{1}{2+x}-\frac{1}{2}}{x}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{1}\boxed{÷}\boxed{(}\boxed{2}\boxed{+}\boxed{\text{X}}\boxed{)}\boxed{-}\boxed{1}\boxed{÷}\boxed{2}\boxed{)}\boxed{÷}\boxed{\text{X}}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(1/\left(2+\text{X}\right)-1/2\right)/\text{X}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

The function $\frac{\frac{1}{2+x}-\frac{1}{2}}{x}$ tends to $-0.25$ as $x$ tends to $0$ from both the sides. So, the value of $\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)$ graphically is $-0.25$ .

Check the value of the limit algebraically.

  $\begin{array}{c}\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)=\underset{x\to 0}{\lim }\left(\frac{\frac{2-\left(2+x\right)}{2\left(2+x\right)}}{x}\right)\\ =\underset{x\to 0}{\lim }\left(\frac{2-2-x}{2x\left(2+x\right)}\right)\\ =\underset{x\to 0}{\lim }\left(-\frac{x}{2x\left(2+x\right)}\right)\\ =\underset{x\to 0}{\lim }\left(-\frac{1}{2\left(2+x\right)}\right)\end{array}$

Further simplify.

  $\begin{array}{c}\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)=-\frac{1}{2\left(2+0\right)}\\ =-\frac{1}{4}\\ =-0.25\end{array}$

So, the value of $\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)$ is equal to $-0.25$ .

Therefore, the value of $\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)$ is $-0.25$ .