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

What are the removable and non-removable discontinuities, if any, of #f(x)=((x-3)(x+5))/((x-3)(x+3))#?

Answer 1

There is a removable discontinuity at #x=3# and a non-removable discontinuity at #x=-3#.

Any rational function is continuous on its domain. Since the domain of this function is all reals except #3# and #-3#, #f# has discontinuities at #3# and #-3#.

#lim_(xrarr3)f(x) = 8/6 = 4/3# so the discontinuity at #3# is removable.

#lim_(xrarr-3)f(x)# does not exist, so the discontinuity at #-3# is non-removable.

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Answer 2

Emilia Aguilar

The function f(x) = ((x-3)(x+5))/((x-3)(x+3)) has a removable discontinuity at x = 3 and a non-removable discontinuity at x = -3.

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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.