Question

the initial substitution of x = a yields the form 0/0. simplify the function algebraically, or use a table or graph to determine the limit. if necessary, state that the limit does not exist. lim(x→2) (2x² + 9x - 26)/(x² - 4) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim(x→2) (2x² + 9x - 26)/(x² - 4) = (type an integer or a simplified fraction.) b. the limit does not exist.

Explanation:

Step1: Factor the expressions

Factor $2x^{2}+9x - 26=(x - 2)(2x+13)$ and $x^{2}-4=(x - 2)(x + 2)$.
So, $\lim_{x
ightarrow2}\frac{2x^{2}+9x - 26}{x^{2}-4}=\lim_{x
ightarrow2}\frac{(x - 2)(2x + 13)}{(x - 2)(x + 2)}$.

Step2: Cancel out common factors

Cancel out $(x - 2)$ (since $x
eq2$ when taking the limit), we get $\lim_{x
ightarrow2}\frac{2x+13}{x + 2}$.

Step3: Substitute $x = 2$

Substitute $x = 2$ into $\frac{2x+13}{x + 2}$, we have $\frac{2\times2+13}{2+2}=\frac{4 + 13}{4}=\frac{17}{4}$.

Answer:

A. $\lim_{x
ightarrow2}\frac{2x^{2}+9x - 26}{x^{2}-4}=\frac{17}{4}$