Question

find lim(x→ - 4) (-3x - 12)/(x³ + 4x²). select the correct choice below and, if necessary, fill in the ans a. lim(x→ - 4) (-3x - 12)/(x³ + 4x²)=-3/16 (type an integer or a simplified fraction.) b. the limit does not exist.

Explanation:

Step1: Factor the numerator and denominator

Factor -3x - 12 as -3(x + 4), and factor \(x^{3}+4x^{2}\) as \(x^{2}(x + 4)\). So the limit becomes \(\lim_{x
ightarrow - 4}\frac{-3(x + 4)}{x^{2}(x + 4)}\).

Step2: Cancel out the common factor

Cancel out the common factor (x + 4) (since \(x
eq - 4\) when taking the limit), we get \(\lim_{x
ightarrow - 4}\frac{-3}{x^{2}}\).

Step3: Substitute x = - 4

Substitute x=-4 into \(\frac{-3}{x^{2}}\), we have \(\frac{-3}{(-4)^{2}}=-\frac{3}{16}\).

Answer:

A. \(\lim_{x
ightarrow - 4}\frac{-3x - 12}{x^{3}+4x^{2}}=-\frac{3}{16}\)