Question

Prove the limit exists if and only if the left and right limits exist and are equal

Collected in the board: Limit

Steven Zheng posted 1 month ago

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.



Steven Zheng posted 1 month ago

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