Calculus Problems and Solutions
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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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Show that as positive real numbers get smaller their inverse gets larger.
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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 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 0}\,\frac{x^2(1+x)}{2x}.\end{equation}
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Determine \begin{equation}\lim_{x\to 1}\,\frac{x}{x+1}.\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}
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Function Limits of Quotients of Polynomials
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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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Pinching Theorem of Function Limits
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Prove that \begin{equation}\lim_{x\to a}\,\frac{1}{f(x)}=\frac{1}{L}.\end{equation}
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Prove the triangle inequality.
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Prove that function limits are unique.
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Function Limits of Polynomials
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Prove that \begin{equation}\lim_{x\to a}\,f(x)\,g(x)=LK.\end{equation}
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Prove that \begin{equation}\lim_{x\to a}\,\sum_{i=1}^N\,f_i(x)=\sum_{i=1}^N\,L_i.\end{equation}
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Prove that \begin{equation}\lim_{x\to a}\,K\,f(x)=KL\end{equation} where $K$ is a constant and \begin{equation}\lim_{x\to a}\,f(x)=L.\end{equation}
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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 \[\lim_{x\to a}\,[f(x)+g(x)]=L+K.\]
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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 $a\,b\leq|a|\,|b|$ for all real numbers $a$ and $b$.
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Prove that $||a|-|b||\leq|a-b|$.
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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}
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Limit of $x^2$ with a discontinuity at $x=3$.
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What is the limit of $\frac{x^3-5x^2+3x-15}{x-5}$ as $x$ approaches 5?
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What is the limit of $\frac{x^3-5x^2+3x-15}{x-5}$ as $x$ approaches 2?
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Derivative of $x^2$
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Determine the derivative of $(f\circ g)(x)$ where $f(x)=2\,x-2\,x^2$ and $g(x)=1-2\,x$.
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Determine the derivative of $(f\circ g)(x)$ where $f(x)=-x^2-8\,x-6$ and $g(x)=9\,x^2+4\,x+3$.
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Determine the derivative of $(f\circ g)(x)$ where $f(x)=5$ and $g(x)=x^2+8\,x+3$.
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Determine the derivative of $(f\circ g)(x)$ where $f(x)=-x-1$ and $g(x)=-3\,x^2$.
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Determine the derivative of $(f\circ g)(x)$ where $f(x)=2\,x^2+4\,x-1$ and $g(x)=3\,x^2+2\,x-3$.
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Determine the derivative of $(f\circ g)(x)$ where $f(x)=-x^2-5$ and $g(x)=x+5$.
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Determine the derivative of $(f\circ g)(x)$ where $f(x)=2\,x^2+4\,x-4$ and $g(x)=-5\,x^2+2\,x+6$.
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Determine the derivative of $(f\circ g)(x)$ where $f(x)=x^2-6\,x-4$ and $g(x)=-3\,x^2-2\,x-7$.
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