What are the removable and non-removable discontinuities, if any, of #f(x)=(x-2)/(x^2 + x - 6)#?

1 Answer
Trevor Ryan.
Jan 5, 2016

#x=2# is a removable discontinuity.
#x=-3# is a non-removable discontinuity (vertical asymptote).

Explanation:

Factorizing the denominator of the function as a trinomial yields

#f(x)=(x-2)/((x+3)(x-2))#

The #(x-2)# factors hence cancel and so #x=2# is a removable discontinuity.

#x=-3# is a non-removable discontinuity and is called a vertical asymptote.
(Reason : division by zero is undefined).

The graph makes it clear :

graph{(x-2)/(x^2+x-6) [-10, 10, -5, 5]}

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