Home > Calculus > Classifying Topics of Discontinuity (removable vs. non-removable)

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

Aurora Alvarado

Use the definition of absolute value to analyze the function.

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.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Christopher Allers

The function f(x) = |x+2|/(x+2) is not continuous at x = -2. This is a removable discontinuity.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.