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#.

#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.