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

What is the removable and nonremovable discontinuities in the equation #(x-2)/(x^2 + x - 6)#?

Answer 1

The discontinuity you can't remove is at #x=-3#, because it is the zero of the denominator only.

But discontinuity at 2 can be removed, because at this point the value is not defined (both numerator and denominator are 0), but the limit #lim_(x->2)f(x)# exists and is equal to #1/5#, so if you define a function:

#g(x)={ (((x-2)/(x^2+x-6)) if x!=2), (1/5 if x=2) :}#

it is still discontinuous at -3, but it is continuous at 2

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Answer 2

Aurora Alvarado

The removable discontinuity occurs when the numerator of the rational function is equal to zero, resulting in a hole in the graph. In the given equation, the removable discontinuity occurs at x = 2.

The nonremovable discontinuity, also known as a vertical asymptote, occurs when the denominator of the rational function is equal to zero. In the given equation, the nonremovable discontinuity occurs at x = -3 and x = 2.

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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.