Question

the initial substitution of x = a yields the form $\frac{0}{0}$. simplify the function algebraically, or use a table or graph to determine the limit. if necessary, state that the limit does not exist.
$lim_{x
ightarrow14}\frac{x^{2}+3x - 238}{x^{2}-196}$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $lim_{x
ightarrow14}\frac{x^{2}+3x - 238}{x^{2}-196}=$ (type an integer or a simplified fraction.)
b. the limit does not exist

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}+3x - 238=(x + 17)(x-14)$ and the denominator $x^{2}-196=(x + 14)(x - 14)$. So the function becomes $\lim_{x
ightarrow14}\frac{(x + 17)(x - 14)}{(x+14)(x - 14)}$.

Step2: Simplify the function

Cancel out the common factor $(x - 14)$ (since $x
eq14$ when taking the limit), we get $\lim_{x
ightarrow14}\frac{x + 17}{x + 14}$.

Step3: Substitute $x = 14$

Substitute $x=14$ into $\frac{x + 17}{x + 14}$, we have $\frac{14+17}{14 + 14}=\frac{31}{28}$.

Answer:

A. $\lim_{x
ightarrow14}\frac{x^{2}+3x - 238}{x^{2}-196}=\frac{31}{28}$