Problems
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Prove that the limit \begin{equation} \lim_{x\to\infty}\,ax^n \end{equation} does not exist for all $n\geq 1$.
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Prove that the limit \begin{equation} \lim_{x\to\infty}\,x^n \end{equation} does not exist for all $n\geq 1$..
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Prove that the limit \begin{equation} \lim_{x\to\infty}\,x \end{equation} does not exist.
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Prove that \begin{equation} \lim_{x\to\infty}\,\frac{1}{x^n}=0. \end{equation}
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Prove that \begin{equation} \lim_{x\to\infty}\,\frac{1}{x^2}=0. \end{equation}
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Prove that if $\lim_{x\to\infty}\,f(x)=L$ and $\lim_{x\to\infty}\,g(x)=K$ then \begin{equation} \lim_{x\to\infty}\,f(x)+g(x)=L+K. \end{equation}
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Prove that \begin{equation} \lim_{x\to\infty}\,C=C \end{equation} where $C$ is a constant.
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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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Prove that if \begin{equation} \lim_{x\to\infty}\,f(x)=L \end{equation} and \begin{equation} \lim_{x\to\infty}\,g(x)=K \end{equation} then \begin{equation} \lim_{x\to\infty}\,f(x)\,g(x)=LK. \end{equation}
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Prove that if \begin{equation} \lim_{x\to\infty}\,f(x)=L \end{equation} and $L\neq 0$ then \begin{equation} \lim_{x\to\infty}\,\frac{1}{f(x)}=\frac{1}{L}. \end{equation}
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Prove that \begin{equation}\lim_{x\to \infty}\,\frac{1}{x}=0.\end{equation}
