Question:
Evaluate \(\lim_{x\rightarrow 0}\)f(x), where { |x|/x, x≠0 0, x=0
Evaluate \(\lim_{x\rightarrow 0}\)f(x), where { |x|/x, x≠0 0, x=0
Updated On: Oct 23, 2023
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Solution and Explanation
The given function is
f(x) ={ \(\frac{|x|}{x}\), x≠0 0, x=0
\(\lim_{x\rightarrow 0^-}\) f(x) = \(\lim_{x\rightarrow 0^-}\)[\(\frac{|x|}{x}\)]
= \(\lim_{x\rightarrow 0}\) (\(\frac{-x}{x}\)) [When x is negative, |x| = -x]
= \(\lim_{x\rightarrow 0}\)(-1)
= -1
\(\lim_{x\rightarrow 0^+}\)f(x) = \(\lim_{x\rightarrow 0^+}\) [\(\frac{|x|}{x}\)]
= \(\lim_{x\rightarrow 0}\) (\(\frac{x}{x}\)) [When x is Positive, |x| = x]
= \(\lim_{x\rightarrow 0}\) (1)
= 1
It is observed that \(\lim_{x\rightarrow 0^-}\) f(x)≠\(\lim_{x\rightarrow 0^+}\)f(x)
Hence, \(\lim_{x\rightarrow 0}\) does not exist.
f(x) ={ \(\frac{|x|}{x}\), x≠0 0, x=0
\(\lim_{x\rightarrow 0^-}\) f(x) = \(\lim_{x\rightarrow 0^-}\)[\(\frac{|x|}{x}\)]
= \(\lim_{x\rightarrow 0}\) (\(\frac{-x}{x}\)) [When x is negative, |x| = -x]
= \(\lim_{x\rightarrow 0}\)(-1)
= -1
\(\lim_{x\rightarrow 0^+}\)f(x) = \(\lim_{x\rightarrow 0^+}\) [\(\frac{|x|}{x}\)]
= \(\lim_{x\rightarrow 0}\) (\(\frac{x}{x}\)) [When x is Positive, |x| = x]
= \(\lim_{x\rightarrow 0}\) (1)
= 1
It is observed that \(\lim_{x\rightarrow 0^-}\) f(x)≠\(\lim_{x\rightarrow 0^+}\)f(x)
Hence, \(\lim_{x\rightarrow 0}\) does not exist.
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