determine if the limit below exists. if it does, compute the limit.\nlim(x→6) (x² + 36)/(x - 6)\nselect the…

determine if the limit below exists. if it does, compute the limit.\nlim(x→6) (x² + 36)/(x - 6)\nselect the correct choice below and fill in any answer boxes in your choice.\na. lim(x→6) (x² + 36)/(x - 6) = (simplify your answer.)\nb. the limit does not exist.

determine if the limit below exists. if it does, compute the limit.\nlim(x→6) (x² + 36)/(x - 6)\nselect the correct choice below and fill in any answer boxes in your choice.\na. lim(x→6) (x² + 36)/(x - 6) = (simplify your answer.)\nb. the limit does not exist.

Answer

Explanation:

Step1: Substitute x = 6

If we directly substitute (x = 6) into (\frac{x^{2}+36}{x - 6}), we get (\frac{6^{2}+36}{6 - 6}=\frac{36 + 36}{0}=\frac{72}{0}), which is undefined.

Step2: Analyze one - sided limits

Let's consider the left - hand limit (\lim_{x\rightarrow6^{-}}\frac{x^{2}+36}{x - 6}) and the right - hand limit (\lim_{x\rightarrow6^{+}}\frac{x^{2}+36}{x - 6}). As (x\rightarrow6^{-}), (x-6\rightarrow0^{-}) (a very small negative number) and (x^{2}+36\rightarrow72). So (\lim_{x\rightarrow6^{-}}\frac{x^{2}+36}{x - 6}=-\infty). As (x\rightarrow6^{+}), (x - 6\rightarrow0^{+}) (a very small positive number) and (x^{2}+36\rightarrow72). So (\lim_{x\rightarrow6^{+}}\frac{x^{2}+36}{x - 6}=\infty). Since the left - hand limit and the right - hand limit are not equal ((-\infty\neq\infty)).

Answer:

B. The limit does not exist.