Prove that \[\lim_{x\to a}\,f(x)=L\] if and only if \[\quad\lim_{h\to 0}\,f(a+h)=L.\]

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Lesson Parent:

Problem:

Prove that \[\lim_{x\to a}\,f(x)=L\]
if and only if
\[\quad\lim_{h\to 0}\,f(a+h)=L.\]

Answer:

It is true that \[\lim_{x\to a}\,f(x)=L\quad\mbox{if and only if}\quad\lim_{h\to 0}\,f(a+h)=L.\]

Solution:

Recall that "if and only if" means that we have to prove two things:

Both of these have been proven here and here, respectively.

Therefore,