Prove \begin{equation}\lim_{x\to a\,}\,f(x)=L\end{equation} if and only if \begin{equation}\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L.\end{equation}

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Lesson Parent: 
Function Limits IV: The Gory Details Of One-Sided Limits
Problem: 

Prove \begin{equation}\lim_{x\to a\,}\,f(x)=L\end{equation} if and only if \begin{equation}\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L.\end{equation}

Answer: 

It is true that \begin{equation}\lim_{x\to a\,}\,f(x)=L\end{equation} if and only if \begin{equation}\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L.\end{equation}

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Solution: 

Recall that when we say "if $A$ then $B$" we symbolically denote this by <Sign in to see all the formulas>. When we say "$A$ if and only if $B$" we symbolically denote this by <Sign in to see all the formulas>. To prove statements like "$A$ if and only if $B$" we need to show that

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Using this notation, we are asked to prove that

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We have already shown that

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and that

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Therefore,

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