12/18\nlim x→5 |x - 5| / x - 5

Explanation:

Step1: Recall absolute - value definition

When (x\gt5), (|x - 5|=x - 5); when (x\lt5), (|x - 5|=-(x - 5)). We need to find left - hand and right - hand limits.

Step2: Calculate right - hand limit ((x\to5^{+}))

For (x\to5^{+}), (x\gt5), so (|x - 5|=x - 5). Then (\lim_{x\to5^{+}}\frac{|x - 5|}{x - 5}=\lim_{x\to5^{+}}\frac{x - 5}{x - 5}=1).

Step3: Calculate left - hand limit ((x\to5^{-}))

For (x\to5^{-}), (x\lt5), so (|x - 5|=-(x - 5)). Then (\lim_{x\to5^{-}}\frac{|x - 5|}{x - 5}=\lim_{x\to5^{-}}\frac{-(x - 5)}{x - 5}=-1).

Step4: Determine the limit

Since (\lim_{x\to5^{-}}\frac{|x - 5|}{x - 5}=-1) and (\lim_{x\to5^{+}}\frac{|x - 5|}{x - 5}=1), and (\lim_{x\to5^{-}}\frac{|x - 5|}{x - 5}\neq\lim_{x\to5^{+}}\frac{|x - 5|}{x - 5}), the limit (\lim_{x\to5}\frac{|x - 5|}{x - 5}) does not exist.

Answer:

The limit does not exist.