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Assume that $\begin{equation} \lim_{x\to\infty}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to\infty}\,g(x)=K \end{equation}$ and that $K\neq 0$ . Prove that $\begin{equation}\lim_{x\to\infty}\,\frac{f(x)}{g(x)}=\frac{L}{K}.\end{equation}$

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Lesson Parent:

Function Limits IX: Horizontal Asymptotes And The Limit Of

$f(x)$ As

Problem:

Assume that
$\begin{equation} \lim_{x\to\infty}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to\infty}\,g(x)=K \end{equation}$
and that $K\neq 0$ . Prove that $\begin{equation}\lim_{x\to\infty}\,\frac{f(x)}{g(x)}=\frac{L}{K}.\end{equation}$

Solution:

Since

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is the product of the two functions

$f(x)$ and <Sign in to see all the formulas>, we know that if the limit of

$f(x)$ and the limit of <Sign in to see all the formulas> exists, then the limit of their products exists and is the product of the two limits. We know by our assumptions that the limit of

$f(x)$ exists. Since the limit of

$g(x)$ exists and is nonzero, we also know that the limit of <Sign in to see all the formulas> exists and is

$1/K$ . Therefore,

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SAMPLE SOLUTION

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