How do you evaluate $|x|/x$ as x approaches 0?

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Answer 1 · 2016-05-06T13:30:11

The limit does not exist.

I assume that the question is "How do you evaluate the limit of $absx/x$ as $xrarr0$ ?"

Recall:

$absx = {(x,"if",x >= 0),(-x,"if",x < 0):}$ .

So,

$absx/x = {(x/x,"if",x > 0),(-x/x,"if",x < 0):}$

$= {(1,"if",x > 0),(-1,"if",x < 0):}$ .

Therefore,

$lim_(xrarr0^-)absx/x = -1$ and $lim_(xrarr0^+)absx/x = 1$

Because the two one sided limits are not equal, the limit $lim_(xrarr0)absx/x$ does not exist.

How do you evaluate |x|/x|x|/x as x approaches 0?