Prove that \begin{equation}\lim_{x\to a}\,f(x)\,g(x)=LK.\end{equation}

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Lesson Parent: 
Function Limits V: Properties of Function Limits
Problem: 

If
\begin{equation}
\lim_{x\to a}\,f(x)=L
\end{equation}
and
\begin{equation}
\lim_{x\to a}\,g(x)=K
\end{equation}
then prove that
\begin{equation}
\lim_{x\to a}\,f(x)\,g(x)=LK.
\end{equation}

Answer: 

It is true that \begin{equation}\lim_{x\to a}\,f(x)\,g(x)=LK.\end{equation}

SAMPLE SOLUTION

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Solution: 

First, we know from our rules of inequalities that

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Because

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we know that there exists a <Sign in to see all the formulas> such that <Sign in to see all the formulas> and thus <Sign in to see all the formulas>. Furthermore, we also know that there exists a <Sign in to see all the formulas> such that <Sign in to see all the formulas>.

Because

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we know that there exists a <Sign in to see all the formulas> such that <Sign in to see all the formulas>.

So now we choose the smallest of the three deltas (<Sign in to see all the formulas>) so that

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Therefore we have shown that given any <Sign in to see all the formulas> there exits a <Sign in to see all the formulas> such that <Sign in to see all the formulas>.

There is one important subtlety that we want to point out. We first pointed out that

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The important thing to note is that we used <Sign in to see all the formulas> and not <Sign in to see all the formulas>. If we had used <Sign in to see all the formulas> then we would have found a <Sign in to see all the formulas> such that <Sign in to see all the formulas>. On the surface this seems fine. However, what if $K=0$? Then <Sign in to see all the formulas> would have been infinite and we could not say <Sign in to see all the formulas> is less than any specific number and therefore not been able to say that <Sign in to see all the formulas> would have been less than any specific number. To keep the proof usable for any value of $K$ (even when it is zero) then we use <Sign in to see all the formulas> which means <Sign in to see all the formulas> would be bounded even for $K=0$.