Question
Prove the limit exists if and only if the left and right limits exist and are equal
Answer
The statement is meant to prove the biconditional statements.
1) If the left and right limits exist and are equal, the limit exists.
2) If the limit exits, the left and right limits exist and are equal.
First, if the left limit exists,
\lim\limits_{x\to a^-}f(x) =L
that is, for any real number ϵ>0, there exists another real number δ>0 such that
if
0 < x-a <δ
(1)
then
0< f(x)-L< ϵ
(2)
Similarly, the right limit exists,
\lim\limits_{x\to a^+}f(x) =L
that is, for any real number ϵ>0, there exists another real number δ>0 such that
if
0 < a-x <δ
(3)
then
0< L-f(x)< ϵ
(4)
Combining (1),(3) and (2), (4), obtains the definition of limit,
this is, for any real number ϵ>0, there exists another real number δ>0 such that
such that if
0<|x−a|<δ
(5)
then
|f(x)−L|<ϵ
(6)
Now the first statement is true.
The second, if the limit exists, (1),(3) and (2), (4) could be obtained by removing the absolute value sign of (5) and (6),
that is, the left and right limits exist and are equal.
Now it's the end of the proof - the limit exists if and only if left and right limits exist and are equal.