find the following limit or state that it does not exist. lim x→ - 2 x² + 5x + 6 / x + 2 select the correct…

find the following limit or state that it does not exist. lim x→ - 2 x² + 5x + 6 / x + 2 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim x→ - 2 x² + 5x + 6 / x + 2 = (type an exact answer.) b. the limit does not exist.

find the following limit or state that it does not exist. lim x→ - 2 x² + 5x + 6 / x + 2 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim x→ - 2 x² + 5x + 6 / x + 2 = (type an exact answer.) b. the limit does not exist.

Answer

Explanation:

Step1: Factor the numerator

Factor $x^{2}+5x + 6$ as $(x + 2)(x+3)$. So the limit becomes $\lim_{x\rightarrow - 2}\frac{(x + 2)(x + 3)}{x + 2}$.

Step2: Simplify the function

Cancel out the common factor $(x + 2)$ (for $x\neq - 2$). The function simplifies to $\lim_{x\rightarrow - 2}(x + 3)$.

Step3: Evaluate the limit

Substitute $x=-2$ into $x + 3$. We get $-2+3=1$.

Answer:

A. $\lim_{x\rightarrow - 2}\frac{x^{2}+5x + 6}{x + 2}=1$