Problems
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Prove that \[ \lim_{x\to a}\frac{1}{x}=\frac{1}{a} \] 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}
