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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

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#

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#

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Answer 2

Ezra Adams

The limit of x / |x| as x approaches 0 is undefined.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.