Question

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?

Answer 1

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`.

`abs(x+2) = {(x+2,"if",x+2 > 0),(-(x+2),"if",x+2 < 0):}`

`= {(x+2,"if",x > -2),(-(x+2),"if",x < -2):}`

Using the above, we can write,

`f(x) = abs(x+2)/(x+2) = {((x+2)/(x+2),"if", x > -2),((-(x+2))/(x+2),"if",x < -2) :}`

`= {(1,"if", x > -2),(-1,"if",x < -2) :}`

For all `x != -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.