# How do you find the x values at which f(x)=abs(x+2)/(x+2) is not continuous, which of the discontinuities are removable?

##### 1 Answer
Oct 24, 2016

Use the definition of absolute value to analyze the function.

#### Explanation:

The function is not defined for $x = - 2$, so it is not continuous at $- 2$.

$\left\mid x + 2 \right\mid = \left\{\begin{matrix}x + 2 & \text{if" & x+2 > 0 \\ -(x+2) & "if} & x + 2 < 0\end{matrix}\right.$

$= \left\{\begin{matrix}x + 2 & \text{if" & x > -2 \\ -(x+2) & "if} & x < - 2\end{matrix}\right.$

Using the above, we can write,

$f \left(x\right) = \frac{\left\mid x + 2 \right\mid}{x + 2} = \left\{\begin{matrix}\frac{x + 2}{x + 2} & \text{if" & x > -2 \\ (-(x+2))/(x+2) & "if} & x < - 2\end{matrix}\right.$

$= \left\{\begin{matrix}1 & \text{if" & x > -2 \\ -1 & "if} & x < - 2\end{matrix}\right.$

For all $x \ne - 2$, the function is continuous since each branch is continuous.

At $x = - 2$, the limits from the left and right are not equal, so the limit does not exist. Therefore, the discontinuity is not removable.