Explanation
We are given the following function:
[tex]g(x)=\frac{x-3}{x^{^2}-9}[/tex]We are required to determine the limit of the function above as it approaches 3.
This is achieved as:
[tex]\begin{gathered} \lim_{x\to3}\frac{x-3}{x^2-9}=\frac{3-3}{3^2-9}=\frac{0}{0} \\ this\text{ }form\text{ }is\text{ }called\text{ }an\text{ }indeterminant\text{ }form \end{gathered}[/tex]Since the method above cannot give an information on the limit, we need to try another method as follows:
[tex]\begin{gathered} \lim_{x\to3}\frac{x-3}{x^2-9}=\lim_{x\to3}\frac{x-3}{x^2-3^2}=\lim_{x\to3}\frac{x-3}{(x-3)(x+3)} \\ =\lim_{x\to3}\frac{1}{x^+3} \\ =\frac{1}{3+3}=\frac{1}{6} \end{gathered}[/tex]Hence, the answer is 1/6.
