Question

calculate the following limit: \\(\lim\limits_{x\to 2} \dfrac{5}{(x - 2)}\\).\
\\(\bigcirc\\) 0\
\\(\bigcirc\\) does not exist\
\\(\bigcirc\\) \\(\infty\\)\
\\(\bigcirc\\) \\(-\infty\\)\
\\(\bigcirc\\) \\(\dfrac{5}{2}\\)

Explanation:

Step1: Analyze the denominator as \(x\to2\)

As \(x\) approaches \(2\), the expression \((x - 2)\) approaches \(0\). So we have a fraction \(\frac{5}{x - 2}\) where the denominator approaches \(0\) and the numerator is a non - zero constant (\(5\)).

Step2: Analyze the left - hand limit (\(x\to2^{-}\))

When \(x\) approaches \(2\) from the left (\(x<2\)), \(x - 2<0\) and \(|x - 2|\) approaches \(0\). So \(\frac{5}{x - 2}=\frac{5}{-|x - 2|}\), and as \(|x - 2|\to0^{+}\), \(\frac{5}{x - 2}\to-\infty\).

Step3: Analyze the right - hand limit (\(x\to2^{+}\))

When \(x\) approaches \(2\) from the right (\(x > 2\)), \(x - 2>0\) and \(|x - 2|\) approaches \(0\). So \(\frac{5}{x - 2}=\frac{5}{|x - 2|}\), and as \(|x - 2|\to0^{+}\), \(\frac{5}{x - 2}\to+\infty\).

Step4: Determine the existence of the limit

Since the left - hand limit (\(\lim_{x\to2^{-}}\frac{5}{x - 2}=-\infty\)) and the right - hand limit (\(\lim_{x\to2^{+}}\frac{5}{x - 2}=+\infty\)) are not equal, the two - sided limit \(\lim_{x\to2}\frac{5}{x - 2}\) does not exist.

Answer:

Does Not Exist