Find the limit (x^2+x-6)/(x+3) when x approach -3.

For evaluating the limit, we'll choose the dividing out
technique.

We'll apply the direct substitution, by
substituting the unknown x, by the value -3 and we'll see that it fails, because both,
numerator and denominator, are cancelling for x=-3. That means x=-3 is a root for both,
that means that (x+3) is a common factor for both.

We'll
write the numerator using the
formula:

x^2+x-6=(x-x1)(x-x2), where x1, x2 are the roots
and
x1=-3

x^2+x-6=(x+3)(x-x2)

We
also know that x1+x2 = -1, -3+x2=-1

and x1*x2=-6,
(-3)*x2=-6

x2=2

Now, we'll
evaluate the limit:

lim (x^2+x-6)/(x+3) = lim
(x+3)(x-2)/(x+3)

Now, we can divide out like
factor:

lim (x^2+x-6)/(x+3) = lim
(x-2)

We can apply the replacement theorem and we'll
get:

lim (x-2) = -3-2 =
-5

So, lim (x^2+x-6)/(x+3) =
-5.