Question

find \\( \lim _{x \
ightarrow 9} \frac{x - 9}{x^{2} - 81} \\).

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. \\( \lim _{x \
ightarrow 9} \frac{x - 9}{x^{2} - 81}= \\)
(type an integer or a simplified fraction.)

b. the limit does not exist.

Explanation:

Step1: Factor the denominator

Use the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\). Here, \(x^{2}-81=(x + 9)(x - 9)\). So, \(\lim_{x
ightarrow9}\frac{x - 9}{x^{2}-81}=\lim_{x
ightarrow9}\frac{x - 9}{(x + 9)(x - 9)}\).

Step2: Simplify the function

Cancel out the common factor \((x - 9)\) (for \(x
eq9\)). The function simplifies to \(\lim_{x
ightarrow9}\frac{1}{x + 9}\).

Step3: Substitute \(x = 9\)

Substitute \(x=9\) into \(\frac{1}{x + 9}\). We get \(\frac{1}{9+9}=\frac{1}{18}\).

Answer:

A. \(\lim_{x
ightarrow9}\frac{x - 9}{x^{2}-81}=\frac{1}{18}\)