evaluate the following limit by simplifying the expression (first answer box) and then evaluating the limit…

evaluate the following limit by simplifying the expression (first answer box) and then evaluating the limit (second answer box). (lim_{x\rightarrow9}left(\frac{\frac{1}{x - 9}-\frac{18}{x^{2}-81}}{}\right)=lim_{x\rightarrow9}square=square). note: in your written solution, you should write the limit statement (lim_{x\rightarrow9}) in every step except the last one, where the limit is finally evaluated. note: you can earn partial credit on this problem.

evaluate the following limit by simplifying the expression (first answer box) and then evaluating the limit (second answer box). (lim_{x\rightarrow9}left(\frac{\frac{1}{x - 9}-\frac{18}{x^{2}-81}}{}\right)=lim_{x\rightarrow9}square=square). note: in your written solution, you should write the limit statement (lim_{x\rightarrow9}) in every step except the last one, where the limit is finally evaluated. note: you can earn partial credit on this problem.

Answer

Explanation:

Step1: Factor the denominator

First, factor (x^{2}-81=(x - 9)(x + 9)). The original limit (\lim_{x\rightarrow9}\left(\frac{1}{x - 9}-\frac{18}{x^{2}-81}\right)=\lim_{x\rightarrow9}\left(\frac{1}{x - 9}-\frac{18}{(x - 9)(x + 9)}\right)).

Step2: Find a common - denominator

The common denominator of the two fractions is ((x - 9)(x + 9)). So (\lim_{x\rightarrow9}\left(\frac{x + 9-18}{(x - 9)(x + 9)}\right)=\lim_{x\rightarrow9}\frac{x - 9}{(x - 9)(x + 9)}).

Step3: Simplify the fraction

Cancel out the common factor ((x - 9)) (since (x\neq9) when taking the limit), we get (\lim_{x\rightarrow9}\frac{1}{x + 9}).

Step4: Evaluate the limit

Substitute (x = 9) into (\frac{1}{x+9}), we have (\frac{1}{9 + 9}=\frac{1}{18}).

Answer:

(\lim_{x\rightarrow9}\left(\frac{1}{x - 9}-\frac{18}{x^{2}-81}\right)=\frac{1}{18})