How do you evaluate the limit of #lim (x+2)/(x^2-4)# as #x->-2#?

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Answer 1 · 2016-12-08T21:03:18

Factor the denominator and cancel like factors or use L'Hôpital's rule ; either way works.

#lim_(xto-2)(x + 2)/(x^2 - 4)#

Factor the denominator:

#lim_(xto-2)(x + 2)/((x - 2)(x + 2))#

Cancel common factors:

#lim_(xto-2)cancel(x + 2)/((x - 2)cancel(x + 2))#

Evaluate at the limit.

#lim_(xto-2)1/(x - 2) = 1/(-2 - 2) = -1/4#

The derivative of the numerator:

#(d(x + 2))/dx = 1#

The derivative of the denominator:

#(d(x^2 - 4))/dx = 2x#

#lim_(xto-2) 1/2x = 1/(2(-2)) = -1/4#