Find the limit as x approaches -4 of (x^2 + 5x + 4)/(x^2 + 3x - 4).

Step-by-step solution

1. 1

   Substitute first to classify the limit. Numerator: $(-4)^2+5(-4)+4=16-20+4=0$. Denominator: $(-4)^2+3(-4)-4=16-12-4=0$. The form is $\tfrac00$, which means a common factor of $(x+4)$ is hiding in both.

2. 2

   Factor the numerator. Two numbers multiplying to $4$ and adding to $5$ are $1$ and $4$: $x^2+5x+4=(x+1)(x+4).$

3. 3

   Factor the denominator. Two numbers multiplying to $-4$ and adding to $3$ are $4$ and $-1$: $x^2+3x-4=(x+4)(x-1).$

4. 4

   Cancel the shared factor. For $x\ne-4$, $\frac{(x+1)(x+4)}{(x+4)(x-1)}=\frac{x+1}{x-1}.$ Cancelling is legitimate because a limit never evaluates the function at the point itself.

5. 5

   Substitute into the reduced form. $\lim_{x\to-4}\frac{x+1}{x-1}=\frac{-4+1}{-4-1}=\frac{-3}{-5}=\frac35.$ Testing $x=-4.001$ and $x=-3.999$ in the original quotient gives $0.60008$ and $0.59992$, bracketing $0.6$ as expected.