delta-epsilon definition of a limit

Question

I was unable to find a method online how to use the δ-ε definition of a limit to prove that the limit is L for a function which includes an absolute value.  

 

My function is

 

  lim     |x-2|

^x→(-2)

 

Thanks,

Mike

Answer 1

Since the function is continuous we can substitute x = -2 to find that the limit is |-2 - 2| = |-4| = 4.

 

The δ-ε proof is as follows.

 

Using the reverse triangle inequality, we have | |x-2| - 4 | < | (x-2) +4 | = |x - (-2)|.

 

Hence, if |x - (-2)| < δ = ε, then | |x-2| - 4 |< ε.