Question
Download Solution PDFThe value of \(\rm \displaystyle\lim_{x\rightarrow 0} \dfrac{|x|}{x}\) is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
For a limit to exist, Left-hand limit and right-hand limit must be equal.
Calculations:
For a limit to exist Left-hand limit and right-hand limit must be equal.
|x| can have two values
|x | = - x when x is negative
|x| = x when x is positive.
\(\rm \displaystyle\lim_{x\rightarrow 0^-} \dfrac{|x|}{x}\) = \(\rm \displaystyle\lim_{x\rightarrow 0} \dfrac{-x}{x} = -1\)
\(\rm \displaystyle\lim_{x\rightarrow 0^+} \dfrac{|x|}{x}\) = \(\rm \displaystyle\lim_{x\rightarrow 0} \dfrac{x}{x} = 1\)
Here, \(\rm \displaystyle\lim_{x\rightarrow 0^-} \dfrac{|x|}{x} \neq \rm \displaystyle\lim_{x\rightarrow 0^+}\dfrac{|x|}{x}\)
Hence, \(\rm \displaystyle\lim_{x\rightarrow 0} \dfrac{|x|}{x}\)does not exist.
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