How do you find the limit #lim_(x->0^-)|x|/x# ?

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. It can be broken down into this:

#|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 this one-sided limit, we must figure out which version of this function is appropriate for our question.

Because our limit is approaching #0# from the negative side, we must use the version of #|x|# that is #<0#, which is #-x#. Rewriting our original problem, we have:

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

Now that the absolute value is gone, we can divide the #x# term and now have:

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

One of the properties of limits is that the limit of a constant is always that constant. If you imagine a constant on a graph, it would be a horizontal line stretching infinitely in both directions, since it stays at the same #y#-value regardless of what the #x#-value does. Since limits deal with finding what value a function "approaches" as it reaches a certain point, the limit of a horizontal line will always be a point along that line, no matter what x-value is chosen. Because of this, we now know:

#lim_(x->0^-)-1 = -1#, Giving us our final answer.