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,