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

Question

Paisley Johnson · Accepted 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|# can be broken down into this:

#|x|=# #x#, when #x≥0# -#x#, when #x<0#

As you can see, the absolute value function's primary use is to return a non-negative number regardless of the value of x that is selected. Therefore, in order to evaluate a one-sided limit, we need to determine which version of this function is suitable for our particular question.

Finding the version of the absolute value function that has negative values near that point is necessary if the limit we are looking for is approaching from the negative side. For instance:

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

If we were to dissect this function into its component parts, we would obtain:

#|2x+4| = # #2x+4#, when #x>=-2# #-(2x+4)#, when #x<-2#

#-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.

Now that we have the right version of the absolute value function in our limit problem, 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).

Adam Adkins · Answer

undefined To find a one-sided limit for an absolute value function, evaluate the function separately for each side of the limit. For the left-hand limit, approach the given value from the left side, and for the right-hand limit, approach from the right side. Substitute the value into the function and simplify the expression.