Question

Write the value of $\underset{x\to 2}{\lim }\frac{\left|x-2\right|}{x-2}.$

Answer 1

$$\begin{gathered} \underset{x\to 2}{\lim }\frac{\left|x-2\right|}{x-2} \\ \mathrm{LHL}: \\ \underset{x\to 2^{-}}{\lim }\left(\frac{\left|x-2\right|}{x-2}\right) \\ \mathrm{Let}x=2-h \\ \mathrm{If}x\to 2,\mathrm{then}h\to 0. \\ =\underset{h\to 0}{\lim }\frac{\left|2-h-2\right|}{2-h-2}=-1 \\ \mathrm{RHL}: \\ \underset{x\to 2^{+}}{\lim }\left(\frac{\left|x-2\right|}{x-2}\right) \\ \mathrm{Let}x=2+h \\ \mathrm{If}x\to 2,\mathrm{then}h\to 0. \\ \underset{h\to 0}{\lim }\frac{\left|2+h-2\right|}{2+h-2}=1 \\ \mathrm{LHL}\ne \mathrm{RHL} \\ \mathrm{Thus},\underset{x\to 2}{\lim }\left(\frac{\left|x-2\right|}{x-2}\right)\mathrm{does}\mathrm{not}\mathrm{exist}. \end{gathered}$$