Problems
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Prove that lim for any a in (-\infty,0).
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Prove that \lim_{x\to a}\frac{1}{x}=\frac{1}{a} for any a in (0,\infty).
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Determine the function limit \lim_{x\to 1}\,\frac{x^2-1}{x^2-2x+1}.
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Prove that \lim_{x\to 3}\,f(x)=9 where f(x)=\left\{\begin{array}{c}3x^2-18&x\neq 3 \\ 5 & x=3\\\end{array}\right.
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Determine the limit \lim_{x\to 2}\,\frac{5x^2-3x-14}{x-2}.
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Prove that \lim_{x\to a}\,f(x)=L if and only if \lim_{x\to a}\,(f(x)-L)=0.
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Prove that if \lim_{x\to a}\,(f(x)-L)=0, then \lim_{x\to a}\,f(x)=L.
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Prove that if \lim_{x\to a}\,f(x)=L, then \lim_{x\to a}\,(f(x)-L)=0.
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Prove that \lim_{h\to 0}\,f(a+h)=L if and only if \lim_{x\to a}\,f(x)=L.
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Prove that \lim_{x\to 4}\,\sqrt{x}=2.
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Prove that \lim_{h\to 0}\,\sqrt{x+h}+\sqrt{x}=2\sqrt{x}.
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Prove that if \lim_{h\to 0}\,g(h)=0 and g(h)\neq 0 for some interval (-\bar{\delta},0)\cup (0,\bar{\delta}) where \bar{\delta}\gt 0 then \lim_{h\to 0}\,g(h)=\lim_{g(h)\to 0}\,g(h).
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Prove that if \lim_{h\to 0}\,f(a+h)=L, then \lim_{x\to a}\,f(x)=L.
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Prove that if \lim_{x\to a}\,f(x)=L, then \lim_{h\to 0}\,f(a+h)=L.
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Prove that \lim_{x\to a}\,f(x)=L if and only if \quad\lim_{h\to 0}\,f(a+h)=L.
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Prove that if \lim_{x\to a}\,g(x)=0 then the limit \lim_{x\to a}\,\frac{1}{g(x)} does not exist.
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Prove that if \lim_{x\to a}\,f(x)=L\neq 0 and \lim_{x\to a}\,g(x)=0 then the limit \lim_{x\to a}\,\frac{f(x)}{g(x)} does not exist.
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Prove that \lim_{x\to a}\,\frac{1}{x-a} does not exist.
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Prove that \begin{equation}\lim_{x\to 0}\frac{1}{x}\end{equation} does not exist.
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Determine \begin{equation}\lim_{x\to 5}\,\frac{2x^3-5x^2+6x-15}{x-5}\end{equation}
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Prove that \begin{equation}\lim_{x\to a}\,f(x)/g(x)=L/K\end{equation} if K\neq 0.
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Prove that \begin{equation}\lim_{x\to a}\,K=K\end{equation} where K is a constant.
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Prove that \begin{equation}\lim_{x\to a}\,x^2=a^2.\end{equation}
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Prove that \begin{equation}\lim_{x\to a}\,x\,g(x)=a\,K. \end{equation}
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Prove that \begin{equation}\lim_{x\to a}\,x^n=a^n.\end{equation}
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Prove that \begin{equation}\lim_{x\to a}\,|x|=|a|.\end{equation}
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Prove that \begin{equation}\lim_{x\to a}\,x=a.\end{equation}
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Prove that the limit of a constant number is that number. Show that \begin{equation} \lim_{x\to a}\,K=K \end{equation} where K is a constant.
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Prove that \begin{equation}\lim_{x\to a}\,(m\,x+b)=m\,a+b \end{equation} whem m is not zero.
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Prove that \begin{equation}\lim_{x\to 2}\,2x-1=3.\end{equation}
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Prove that \begin{equation} \lim_{x\to 0}\,4x+2=2. \end{equation}
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Prove that \begin{equation} \lim_{x\to 0}\,4x+2\neq 6. \end{equation}
