calculus limits

Question

Is there a number a such that lim x→−2 (3x² + ax + a + 3) / (x² + x − 2) exists?

If so, find the value of a and the value of the limit.

We are not allowed to use L’hospital’s rule.

Answer 1

Let x = -2 and evaluate the numerator. We get 3x² + ax + a + 3 = 3 * 4 -2a + a + 3 = 15 - a

Set this equal to 0.

Thus, a = 15

Now, we have (3x² + ax + a + 3)/ (x² + x − 2) =

(3x² + 15x + 15 + 3)/ (x² + x − 2) =

(3x² + 15x + 18)/ (x² + x − 2) =

3(x² + 5x + 6)/ (x² + x − 2) =

3(x + 3)(x +2)/ ((x + 2)(x - 1) )

Then, lim x -> -2 3(x + 3)(x +2)/ ((x + 2)(x - 1) ) =

lim x -> -2 3(x + 3)/ ((x - 1) ) =

3(-2+3)/(-2-1) =

-1

Answer 2