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 $x$ Tends To Infinity
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}

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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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