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

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Lesson Parent: 
Function Limits III: The Gory Details
Problem: 

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

Answer: 

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

SAMPLE SOLUTION

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Solution: 

To prove this we need to show the following two things:

The proof is complete.