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|## 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 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## is used for checking the value of ##x## because that is the value where the function switches from positive to negative. Any number above ##-2## will return a positive number and any number below would be negative, and therefore need to have its sign swapped to always return a non-negative number.
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 ##x=-2##, we have:
##lim_(x->-2^-) -2x-4####=-2(-2)-4 ##
##= 4-4 = 0##
Note that if a number besides ##-2## was used for the limit, such as:
##lim_(x->3^+) |2x+4|##
We would still check the piece-wise function to see that ##3 > -2##, but not have to worry about the limit being one-sided. This is because the one-sided aspect of a limit for piece-wise functions only becomes important around the values where it will switch signs or functions (##x=-2## in our case).
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