How do you find the limit of $\frac{| x |}{x}$ $|x|/x$ as x approaches $0$ $0$ ?

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Answer 1 · 2016-08-26T11:11:24

Does not exist

For $x < 0$ $x < 0$ , $\frac{| x |}{x} = \frac{- x}{x} = - 1$ $(abs x)/x = (-x)/x = -1$

For $x > 0$ $x >0$ , $\frac{| x |}{x} = \frac{x}{x} = 1$ $(abs x)/x = x/x = 1$

Thus
$lim_(x to 0^-) abs x/x = -1$

$lim_(x to 0^+) abs x/x = 1$

So the limit does not exist.

graph{|x|/x [-10, 10, -5, 5]}

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