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Prove that $\begin{equation}\lim_{x\to a}\,K\,f(x)=KL\end{equation}$ where $K$ is a constant and $\begin{equation}\lim_{x\to a}\,f(x)=L.\end{equation}$

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Lesson Parent:

Function Limits V: Properties of Function Limits

Problem:

Prove that $\begin{equation}\lim_{x\to a}\,K\,f(x)=KL\end{equation}$ where $K$ is a constant and $\begin{equation}\lim_{x\to a}\,f(x)=L.\end{equation}$

Answer:

It is true that $\begin{equation}\lim_{x\to a}\,K\,f(x)=KL\end{equation}$ where $K$ is a constant and $\begin{equation}\lim_{x\to a}\,f(x)=L.\end{equation}$

Solution:

First let's assume that <Sign in to see all the formulas>. Because

<Sign in to see all the formulas>

I can find a <Sign in to see all the formulas> such that <Sign in to see all the formulas> for every <Sign in to see all the formulas>. Now assuming that we are at some value of

$x$ such that <Sign in to see all the formulas> then

<Sign in to see all the formulas>

and therefore

<Sign in to see all the formulas>

for <Sign in to see all the formulas>.

Now let's assume that $K=0$ . Since <Sign in to see all the formulas> is a constant (of zero) we know that

<Sign in to see all the formulas>

Therefore

<Sign in to see all the formulas>

for all

$K$ .

As an aside, we can say that we can distribute $K$ into the function limit:

<Sign in to see all the formulas>

and therefore

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