Question

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

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 \(\lim_{x
ightarrow6^{-}}\frac{x^{2}+36}{x - 6}\) and the right - hand limit \(\lim_{x
ightarrow6^{+}}\frac{x^{2}+36}{x - 6}\).
As \(x
ightarrow6^{-}\), \(x-6
ightarrow0^{-}\) (a very small negative number) and \(x^{2}+36
ightarrow72\). So \(\lim_{x
ightarrow6^{-}}\frac{x^{2}+36}{x - 6}=-\infty\).
As \(x
ightarrow6^{+}\), \(x - 6
ightarrow0^{+}\) (a very small positive number) and \(x^{2}+36
ightarrow72\). So \(\lim_{x
ightarrow6^{+}}\frac{x^{2}+36}{x - 6}=\infty\).
Since the left - hand limit and the right - hand limit are not equal (\(-\infty
eq\infty\)).

Answer:

B. The limit does not exist.