Problems
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Prove that \[ \lim_{x\to a^-}\,[f(x)+g(x)]=L+K \] if \[ \lim_{x\to a^-}\,f(x)=L \] and \[ \lim_{x\to a^-}\,g(x)=K. \]
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Prove that \[ \lim_{x\to a^+}\,[f(x)+g(x)]=L+K \] if \[ \lim_{x\to a^+}\,f(x)=L \] and \[ \lim_{x\to a^+}\,g(x)=K. \]
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Prove that if \begin{equation}\lim_{x\to a-}\,f(x)\neq \lim_{x\to a+}\,f(x)\end{equation} then \begin{equation}\lim_{x\to a\,}\,f(x)\end{equation} does not exist.
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Determine \begin{equation}\lim_{x\to 1}\,f(x)\quad\mbox{where}\quad f(x)=\left\{\begin{array}{lr}x^2,&x\leq 1\\5x,&x\gt 1\end{array}\right.\end{equation}
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Prove \begin{equation}\lim_{x\to a\,}\,f(x)=L\end{equation} if and only if \begin{equation}\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L.\end{equation}
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Prove that if \begin{equation}\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L\end{equation} then \begin{equation}\lim_{x\to a\,}\,f(x)=L.\end{equation}
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Prove that if \begin{equation}\lim_{x\to a\,}\,f(x)=L\end{equation} then \begin{equation}\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L.\end{equation}
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Determine \begin{equation}\lim_{x\to 2}\,f(x)\quad\mbox{where}\quad f(x)=\left\{\begin{array}{lr}x^2,&x\leq 1\\5x,&x\gt 1\end{array}\right.\end{equation}
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Determine \begin{equation}\lim_{x\to 3}\,f(x)\end{equation} where \begin{equation}f(x)=\left\{\begin{array}{lr}x^2,&x\lt3\\10,&x=3\\2x+3,&x\gt 3\end{array}\right.\end{equation}
