Question

How do you find the limit of `x / |x|` as x approaches `0`?

Answer 1

The limit depends upon which side of `0` that `x` approaches from

Explanation:

If `x` is negative but approaching 0
`color(white)("XXX")`written `lim_(xrarr0^-)`
then
since `x/abs(x)=-1` for all negative values of `x`
`color(white)("XXX")lim_(xrarr0^-) x/abs(x)=-1`

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If `x` is positive but approaching 0
`color(white)("XXX")`written `lim_(xrarr0^+)`
then
since `x/abs(x)=+1` for all positive values of `x`
`color(white)("XXX")lim_(xrarr0^+) x/abs(x)=1`