evaluate the limit, if it exists. (if an answer does not exist, enter \nlim(x→ - 4) (3x^2 + 11x - 4)/(x^2…

Explanation:

Step1: Factor the numerator and denominator

The numerator $3x^{2}+11x - 4=(3x - 1)(x + 4)$ and the denominator $x^{2}-16=(x + 4)(x - 4)$ using the formulas for factoring quadratic expressions $ax^{2}+bx + c$ and $a^{2}-b^{2}=(a + b)(a - b)$. So the limit becomes $\lim_{x\rightarrow - 4}\frac{(3x - 1)(x + 4)}{(x + 4)(x - 4)}$.

Step2: Cancel out the common factor

Since $x\neq - 4$ when taking the limit, we can cancel out the common factor $(x + 4)$ in the numerator and denominator. The expression simplifies to $\lim_{x\rightarrow - 4}\frac{3x - 1}{x - 4}$.

Step3: Substitute $x=-4$

Substitute $x = - 4$ into $\frac{3x - 1}{x - 4}$. We get $\frac{3\times(-4)-1}{-4 - 4}=\frac{-12-1}{-8}=\frac{-13}{-8}=\frac{13}{8}$.

Answer:

$\frac{13}{8}$