use algebra and the properties of limits as needed to find the given limit. if the limit does not exist, say…

use algebra and the properties of limits as needed to find the given limit. if the limit does not exist, say so. lim(x→6) (x² - 2x - 24)/(x - 6) select the correct choice and, if necessary, fill in the answer box to complete your choice. a. lim(x→6) (x² - 2x - 24)/(x - 6) = (type an integer or a simplified fraction.) b. the limit does not exist.

use algebra and the properties of limits as needed to find the given limit. if the limit does not exist, say so. lim(x→6) (x² - 2x - 24)/(x - 6) select the correct choice and, if necessary, fill in the answer box to complete your choice. a. lim(x→6) (x² - 2x - 24)/(x - 6) = (type an integer or a simplified fraction.) b. the limit does not exist.

Answer

Explanation:

Step1: Factor the numerator

Factor $x^{2}-2x - 24$. We need two numbers that multiply to - 24 and add up to - 2. These numbers are - 6 and 4. So $x^{2}-2x - 24=(x - 6)(x + 4)$.

Step2: Simplify the function

The original limit $\lim_{x\rightarrow6}\frac{x^{2}-2x - 24}{x - 6}=\lim_{x\rightarrow6}\frac{(x - 6)(x + 4)}{x - 6}$. Cancel out the common factor $(x - 6)$ (since $x\neq6$ when taking the limit), we get $\lim_{x\rightarrow6}(x + 4)$.

Step3: Evaluate the limit

Substitute $x = 6$ into $x+4$. So $\lim_{x\rightarrow6}(x + 4)=6 + 4=10$.

Answer:

A. $\lim_{x\rightarrow6}\frac{x^{2}-2x - 24}{x - 6}=10$