find \\( \\lim _{x \\rightarrow 9} \\frac{x - 9}{x^{2} - 81} \\).\n\nselect the correct choice below and, if…

find \\( \\lim _{x \\rightarrow 9} \\frac{x - 9}{x^{2} - 81} \\).\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. \\( \\lim _{x \\rightarrow 9} \\frac{x - 9}{x^{2} - 81}= \\)\n(type an integer or a simplified fraction.)\n\nb. the limit does not exist.

find \\( \\lim _{x \\rightarrow 9} \\frac{x - 9}{x^{2} - 81} \\).\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. \\( \\lim _{x \\rightarrow 9} \\frac{x - 9}{x^{2} - 81}= \\)\n(type an integer or a simplified fraction.)\n\nb. the limit does not exist.

Answer

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\rightarrow9}\frac{x - 9}{x^{2}-81}=\lim_{x\rightarrow9}\frac{x - 9}{(x + 9)(x - 9)}).

Step2: Simplify the function

Cancel out the common factor ((x - 9)) (for (x\neq9)). The function simplifies to (\lim_{x\rightarrow9}\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\rightarrow9}\frac{x - 9}{x^{2}-81}=\frac{1}{18})