How do you find the limit of (x^2+x-12)/(x-2) as x->2?

1 Answer
Oct 12, 2016

lim_(xrarr2) (x^2+x-12)/(x-2) = color(green)(+-oo)

Explanation:

At x=2
color(white)("XXX")the numerator becomes 2^2+2-12= -6
and
color(white)("XXX")the denominator becomes 2-2=0

L'Hospital's (or L'Hopital's) Rule does not apply. L'Hopital's Rule only applies if both the numerator and denominator are 0.

The limit of a non-negative numerator by a denominator that is approaching 0 is +-oo (depending upon which side the denominator is approaching "0" from).

Here is a graph of the (x^2+x-12)/(x-2).
Note that as xrarr2 the limit does not approach 5 (as implied by those attempting to use L'Hopital's Rule).

enter image source here