Chapter 2, Problem 5RE

To determine

To find: The value of given limit.

Answer to Problem 5RE

The value of $\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)$ is $\frac{-1}{4}$ .

Explanation of Solution

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

Calculation:

Simplify the given expression.

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

To find the limit, substitute 0 for $x$ .

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

Therefore, the value of $\underset{x\to 0}{\lim }\left(\frac{\frac{1}{2+x}-\frac{1}{2}}{x}\right)$ is $\frac{-1}{4}$ .