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

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Lesson Parent:

Problem:

Prove that if
$\begin{equation} \lim_{x\to a}\,g(x)=K \end{equation}$
then
$\begin{equation} \lim_{x\to a} x\,g(x)=a\,K. \end{equation}$

Answer:

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

Solution:

First, we know that

Because

Because

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

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

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

$|x-a|$ would have been unbounded. To keep the proof usable for any value of

$K=0$ .