Find the constant and limit

Question

Here is the limit

$$\lim_{x\rightarrow-2} \frac{4x^2 + ax + a + 12}{x^2 + x - 2}.$$

a) Find the constant $a$

b) find the limit

I dont think it is solvable because it didn't tell me as $x \rightarrow -2$, $y \rightarrow $?

Answer 1

I think it's saying that if the limit exists, find $a$ and find the limit.

If $x \to -2$ then the denominator $\to 0$, so in order for the limit to exist, then the numerator must also $\to 0$.

$(4x^2 +ax +a+12) \to (16 -2a +a+12) = (28-a)$ therefore $a = 28$ So that expression becomes:

$$\lim_{x \to -2} \frac{4x^2 + 28x +40}{x^2+x-2} = \lim_{x \to -2} \frac{4(x+2)(x+5)}{(x+2)(x-1)} = -4$$