- #1
RPierre
- 10
- 0
Homework Statement
Prove that the limit as x->c of f(x) = L if and only if both one sided limits also = L
Homework Equations
Has to be an epsilon delta proof
The Attempt at a Solution
Being an if and only if, I have to do two cases : If A, then B. and if NOT A, then NOT B, logically.
Case 1:
Let lim x->c from the left be L, and lim x->c from the right be L.
then if [tex] c - \delta < x < c then |f(x) - L| < \epsilon [/tex]
and if [tex] c < x < c + \delta then |f(x) - L| < \epsilon [/tex]
Case 2:
Let lim x->c from the left = M, and lim x->c from the right = N.
This is all I have really rationalized I am kind of stumped how to do a rigorous proof of this, I.e. I know how to do specific proofs but not a rigorous general proof. \
Can anyone offer any help / a starting point =/ ?