What is the limit of f(x)=x/|x| as x approaches 0?

Question

#lim f(x)=x/|x| x->0 #

2102 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-01-24T08:19:13

Undefined.

#lim_(x->0)(x/(|x|))#

In order to solve this, you need to know the following about the absolute value of a variable.

The absolute value of #x->|x|=xcolor(white)(888888)# If and only if #x>=0#

The absolute value of #x->|x|=-xcolor(white)(888)# If and only if #<0#

Testing both right and left hand limits:

Right Hand

#x# is positive, so #|x|=x#

#lim_(x->0^+)(x/(x))=>(1/1)=1#

Left Hand

#x# is negative, so #|x|=-x#

#lim_(x->0^-)(x/(-x))=>(-1/1)=-1#

#lim_(x->0^+)(x/(|x|))!=lim_(x->0^-)(x/(|x|))#

#:.#

#lim_(x->0)(x/(|x|))# Undefined