Question

find the following limit or state that it does not exist. \\(\lim\limits_{x \to 18} \frac{x^2 - 324}{18 - x}\\) select the correct choice below and, if necessary, fill in the answer box to complete your choice. \\(\boldsymbol{\text{a. } \lim\limits_{x \to 18} \frac{x^2 - 324}{18 - x} = \square}\\) (type an exact answer.) \\(\boldsymbol{\text{b. the limit does not exist.}}\\)

Explanation:

Step1: Factor the numerator

We know that \(x^2 - 324\) is a difference of squares, so \(x^2 - 324=(x - 18)(x + 18)\). Also, note that \(18 - x=-(x - 18)\). So we can rewrite the function as:
\(\frac{x^2 - 324}{18 - x}=\frac{(x - 18)(x + 18)}{-(x - 18)}\)

Step2: Cancel the common factor

For \(x
eq18\) (since we are taking the limit as \(x
ightarrow18\), we can consider \(x\) close to 18 but not equal to 18), we can cancel out the common factor \((x - 18)\) from the numerator and the denominator. After canceling, we get:
\(\frac{(x - 18)(x + 18)}{-(x - 18)}=- (x + 18)\)

Step3: Evaluate the limit

Now we find the limit as \(x
ightarrow18\) of \(-(x + 18)\). We substitute \(x = 18\) into the function \(-(x + 18)\):
\(\lim_{x
ightarrow18}-(x + 18)=-(18 + 18)=- 36\)

Answer:

\(-36\)