$\underset{x→5}{\lim}\frac{x-5}{|x-5|}$ equals

none of these

$\underset{x→5^+}{\lim}\frac{x-5}{|x-5|}=\underset{x→5^+}{\lim}1=1$,

because for values to the right of  5, x – 5 > 0 ⇒ |x - 5| = (x – 5).

$\underset{x→5^-}{\lim}\frac{x-5}{|x-5|}=\underset{x→5^-}{\lim}-1=-1$

because for values to the left of 5, x – 5 > 0 ⇒ |x - 5| = -(x – 5).

$⇒\underset{x→5}{\lim}\frac{x-5}{|x-5|}$ doesn’t exist

Hence (D) is the correct answer.