Question

Evaluate : $\underset{x\to a}{lim}\frac{x-a}{|x-a|}$

• A. $a$
• B. $0$
• C. $-a$
• D. Does not exist

Answer 1

The correct option is D Does not exist

Given : $\underset{x\to a}{lim}\frac{x-a}{|x-a|}$

Considering $RHL=\underset{x\to a}{lim}\frac{x-a}{|x-a|}=\underset{x\to a^{+}}{lim}\frac{x-a}{x-a}=1$

and now, $LHL=\underset{x\to a^{-}}{lim}\frac{x-a}{|x-a|}=\underset{x\to a^{-}}{lim}\frac{x-a}{-(x-a)}=-1$

Here $RHL\ne LHL$ Therefore, limit does not exist.