# How do you find a one sided limit for an absolute value function?

##### 1 Answer

When dealing with one-sided limits that involve the absolute value of something, the key is to remember that the absolute value function is really a piece-wise function in disguise. For example,

#|x|=#

#x# , when#x≥0#

-#x# , when#x<0#

You can see that no matter what value of x is chosen, it will always return a non-negative number, which is the main use of the absolute value function. This means that to evaluate a one-sided limit, we must figure out which version of this function is appropriate for our question.

If the limit we are trying to find is approaching from the negative side, we must find the version of the absolute value function that contains negative values around that point, for example:

#lim_(x->-2^-) |2x+4|#

If we were to break this function down into its piece-wise form, we would have:

#|2x+4| = #

#2x+4# , when#x>=-2#

#-(2x+4)# , when#x<-2#

If we now replace the absolute value function in our limit problem with the correct version, we would have:

#lim_(x->-2^-) -(2x+4) = lim_(x->-2^-) -2x-4#

Substituting

#lim_(x->-2^-) -2x-4# #=-2(-2)-4 #

#= 4-4 = 0#

Note that if a number besides

#lim_(x->3^+) |2x+4|#

We would still check the piece-wise function to see that