How to apply l'Hopital to calculate the limit of the function (x^2-x-6)/(x+2) x-->-2

To calculate the limit, we'll substitute x by -2 in the
expression of the function.

lim f(x) = lim ( 4 + 2 -6
)/( -2 + 2 )

lim ( x^2 - x -6 )/( x+2 ) =  (6-6)/(-2+2) =
0/0

We notice that we've obtained an indetermination case,
0/0 type.

We'll apply l'Hospital's rule, by differentiating
separately the numerator and denominator.

We'll
differentiate the numerator:

( x^2 - x -6)' = 2x -
1

We'll differentiate the
denominator:

(x+2)' = 1

We'll
re-write the limit:

lim ( x^2 - x -6 )/( x+ 2 ) = lim
(2x-1)

We'll substitute x by
-2:

lim (2x-1) = 2*(-2) - 1 =
-4-1

The limit of the function, when
x->-2, is: lim (x^2-x-6)/(x+2) =
-5.