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1. \(\lim\limits_{x\rightarrow 4}\,(-3)=\)_____;
2. \(\lim\limits_{x\rightarrow 0}\pi =\) _____

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\(\lim\limits_{x\rightarrow c}f(x)=3\), then \(\lim\limits_{x\rightarrow c}[f(x)]^{5}=\) _____.

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If \(\lim\limits_{x\rightarrow c}f(x)=64\), then \(\lim\limits_{x\rightarrow c}\sqrt[3]{f(x)}=\) _____.

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1. \(\lim\limits_{x\rightarrow -1}x\) = _____;
2. \(\lim\limits_{x\rightarrow e}x=\) _____

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1. \(\lim\limits_{x\rightarrow 0}\,( x-2)\) = _____;
2. \(\lim\limits_{x\rightarrow {1}/{2}} (3+x)\) =_____

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1. \(\lim\limits_{x\rightarrow 2}\, ( -3x)\) = _____;
2. \(\lim\limits_{x\rightarrow 0}\, ( 3x)\) = _____

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True or False If \(p\) is a polynomial function, then \(\lim\limits_{x\rightarrow 5}p( x) =p(5) \).

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If the domain of a rational function \(R\) is \(\{x|\) \(x\neq 0\}\), then \(\lim\limits_{x\rightarrow 2}R(x)=R\) (_____).

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True or False Properties of limits cannot be used for one-sided limits.

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True or False If \(f(x)=\dfrac{( x+1) ( x+2) }{x+1}\) and \(g(x)=x+2\), then \(\lim\limits_{x\rightarrow -1}f(x)=\lim\limits_{x\rightarrow -1}g(x)\).

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\({\lim\limits_{x\rightarrow 3}}\, [ 2({x+4})]\)

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\({\lim\limits_{x\rightarrow -2}} [ 3({x+1})]\)

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\({\lim\limits_{x\rightarrow -2}}\, [ x ( 3x{-1}) ( {x+2}) ]\)

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\({\lim\limits_{x\rightarrow -1}} [ x( x-1) ( x+10)] \)

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\({\lim\limits_{t\rightarrow 1}}\,( 3t-2) ^{3}\)

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\({\lim\limits_{x\rightarrow 0}}\,( {-3x+1}) ^{2}\)

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\({\lim\limits_{x\rightarrow 4}} \, (3\sqrt{x})\)

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\({\lim\limits_{x\rightarrow 8}}\left(\dfrac{1}{4}\sqrt[3]{{x}}\right)\)

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\({\lim\limits_{x\rightarrow 3}}\,\sqrt{5x-4}\)

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\({\lim\limits_{t\rightarrow 2}}\,\sqrt{3t+4}\)

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\({\lim\limits_{t\rightarrow 2}} \, [ t\sqrt{( 5t+3) ( t+4 ) }] \)

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\({\lim\limits_{t\rightarrow -1}}\, [ t\sqrt[3]{( {t+1}) ( 2t-1) }] \)

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\({\lim\limits_{x\rightarrow 3}}\, (\sqrt{x}+x+4) ^{1/2}\) \(\ \ \ \ \ \ \qquad \)

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\({\lim\limits_{t\rightarrow 2}}\, (t\sqrt{2t}+4) ^{1/3}\)

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\({\lim\limits_{t\rightarrow -1}}[4t ( t+1) ] ^{2/3}\)

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\(\lim\limits_{x\rightarrow 0}\, ( x^{2}-2x) ^{3/5}\)

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\({\lim\limits_{t\rightarrow 1}} \,( {3t^{2}-2t+4})\)

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\({\lim\limits_{x\rightarrow 0}}\,( {-3x^{4}+2x+1}) \)

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\({\lim\limits_{x\rightarrow \frac{1}{2}}} ( {2x^{4}-8x^{3}+4x-5} ) \)

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\({\lim\limits_{x\rightarrow -\frac{1}{3}}} ( {27x^{3}+9x+1}) \)

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\({\lim\limits_{x\rightarrow 4}}\,{\dfrac{{x^{2}+4}}{\sqrt{{x}}}}\)

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\({\lim\limits_{x\rightarrow 3}}\,{\dfrac{{x^{2}+5}}{\sqrt{{3x}}}}\)

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\({\lim\limits_{x\rightarrow -2}}{\dfrac{{2x^{3}+5x}}{{3x-2}}}\)

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\({\lim\limits_{x\rightarrow 1}}\, {\dfrac{{2x^{4}-1}}{{3x^{3}+2}}}\)

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\({\lim\limits_{x\rightarrow 2}}\,{\dfrac{{x^{2}-4}}{{x-2}}}\)

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\({\lim\limits_{x\rightarrow -2}}\,{\dfrac{{x+2}}{{x^{2}-4}}}\)

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\({\lim\limits_{x\rightarrow -1}}{\dfrac{{x^{3}-x}}{{x+1}}}\)

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\({\lim\limits_{x\rightarrow -1}}\,{\dfrac{{x^{3}+x^{2}}}{{x^{2}-1}} }\)

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\({\lim\limits_{x\rightarrow -8}}\left( {{\dfrac{{2x}}{{x+8}}}+{\dfrac{{16}}{{x+8}}}}\right)\)

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\({ \lim\limits_{x\rightarrow 2}}\left( {{\dfrac{{3x}}{{x-2}}}-{\dfrac{{6}}{{x-2}}}}\right) \)

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\({\lim\limits_{x\rightarrow 2}}\,{\dfrac{{\sqrt{x}-\sqrt{2}}}{{x-2}}\qquad }\)

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\({\lim\limits_{x\rightarrow 3}}\,\dfrac{\sqrt{x}-\sqrt{3}}{x-3}\)

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\({\lim\limits_{x\rightarrow 4}}\,{\dfrac{\sqrt{x+5}-3}{ ( {x-4} ) ( x+1) }}\)

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\({\lim\limits_{x\rightarrow 3}}\,{\dfrac{{\sqrt{x+1}-2}}{x(x-3) }}\)

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\({\lim\limits_{x\rightarrow 3^{-}}} ( {x^{2}-4} ) \)

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\({\lim\limits_{x\rightarrow 2^{+}}} ( {3x^{2}+x}) \)

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\({\lim\limits_{x\rightarrow 3^{-}}}{\dfrac{{x^{2}-9}}{{x-3}}}\)

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\({\lim\limits_{x\rightarrow 3^{+}}}{\dfrac{{x^{2}-9}}{{x-3}}}\)

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\({\lim\limits_{x\rightarrow 3^{-}}}( {\sqrt{9-x^{2}}+x}) ^{2}\)

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\({\lim\limits_{x\rightarrow 2^{+}}}( {2\sqrt{x^{2}-4}+3x}) \)

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\({\lim\limits_{x\rightarrow c}}\, [ {{f(x) -3g}}( x) ] \)

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\({\lim\limits_{x\rightarrow c}}\, [ {5{f(x) }} ] \)

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\({\lim\limits_{x\rightarrow c}}\,{[ {g ( {x}) }] }^{3}\)

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\({\lim\limits_{x\rightarrow c}}\,\dfrac{{f(x) }}{{g(x) -h ( {x}) }} \)

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\({\lim\limits_{x\rightarrow c}}\,\dfrac{h( x) }{g( x) }\)

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\({\lim\limits_{x\rightarrow c}}\,[ 4f( x) \cdot g( x) ] \)

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\({\lim\limits_{x\rightarrow c}}\left[\dfrac{1}{g( x) }\right] ^{2}\)

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\({\lim\limits_{x\rightarrow c}}\,\sqrt[3]{5g( x) -3}\)

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1. \(\lim\limits_{x\rightarrow 4}\left[ f( x) +g( x) \right] \)
2. \(\lim\limits_{x\rightarrow 4}\left\{ f( x) \left[g( x) -h( x) \right] \right\} \)
3. \(\lim\limits_{x\rightarrow 4}[f( x) \cdot g( x)] \)
4. \(\lim\limits_{x\rightarrow 4}[2h( x) ] \)
5. \(\lim\limits_{x\rightarrow 4}\dfrac{g( x) }{f( x) }\)
6. \(\lim\limits_{x\rightarrow 4}\dfrac{h( x) }{f( x) }\)

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1. \(\lim\limits_{x\rightarrow 3}\left\{ 2\left[ f( x) +h( x) \right] \right\} \)
2. \(\lim\limits_{x\rightarrow {3^-}}[ g( x) +h( x) ] \)
3. \(\lim\limits_{x\rightarrow 3}\sqrt[3]{h(x)}\)
4. \(\lim\limits_{x\rightarrow 3}\dfrac{f( x) }{h( x) }\)
5. \(\lim\limits_{x\rightarrow 3}[ h( x) ] ^{3}\)
6. \(\lim\limits_{x\rightarrow 3}[ f( x) -2h( x) ] ^{3/2}\)

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\(f( x) =3x^{2},\) \(c=1\)

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\(f( x) =8x^{3},\) \(c=2\)

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\(f( x) =-2x^{2}+4\), \(c=1\)

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\(f( x) =20-0.8x^{2}\), \(c=3\)

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\(f( x) =\sqrt{x},\) \(c=1\)

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\(f( x) =\sqrt{2x},\) \(c=5\)

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\(f(x)\) \(=4x-3 \)

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\(f(x)=3x+5 \)

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\(f(x)=3x^{2}+4x+1\)

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\(f(x)=2x^{2}+x\)

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\(f(x) =\dfrac{2}{x}\)

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\(f(x) =\dfrac{3}{x^{2}}\)

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\(f(x) ={\left\{ \begin{array}{c@{ }rl} 2x-3 & \hbox{if} & x\leq 1 \\[3pt] 3-x & \hbox{if} & x>1 \end{array} \right. }\) at \(c=1\)

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\(f(x) ={\left\{ \begin{array}{c@{ }rl} 5x+2 & \hbox{if} & x\lt 2 \\[3pt] 1+3x & \hbox{if} & x\geq 2 \end{array} \right. }\) at \(c=2\)

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\(f(x) ={\left\{ \begin{array}{c@{ }rl} 3x-1 & \hbox{if} & x\lt 1 \\[3pt] 4 & \hbox{if} & x=1 \\[3pt] 2x & \hbox{if} & x>1 \end{array} \right. }\) at \(c=1\)

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\(f(x) ={\left\{ \begin{array}{c@{ }rl} 3x-1 & \hbox{if} & x\lt 1 \\[3pt] 2 & \hbox{if} & x=1 \\[3pt] 2x & \hbox{if} & x>1 \end{array} \right. }\) at \(c=1\)

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\(f(x) ={\left\{ \begin{array}{c@{ }rl} x-1 & \hbox{if} & x\lt 1 \\[3pt] \sqrt{\,x-1} & \hbox{if} & x>1 \end{array} \right. }\) at \(c=1\)

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\(f(x) ={\left\{ \begin{array}{c@{ }rl} \sqrt{9-x^{2}} & \hbox{if} & 0 \lt x\lt 3 \\[3pt] \sqrt{x^{2}-9} & \hbox{if} & x>3 \end{array} \right. }\) at \(c=3\)

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\(f(x) ={\left\{ \begin{array}{c@{ }rl} \dfrac{x^{2}-9}{x-3} & \hbox{if} & x\neq 3 \\ 6 & \hbox{if} & x=3 \end{array} \right. }\) at \(c=3\)

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\(f(x) = {\left\{ \begin{array}{c@{ }rl} \dfrac{x-2}{x^{2}-4} & \hbox{if} & x\neq 2 \\ 1 & \hbox{if} & x=2 \end{array} \right. }\) at \(c=2\)

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\(u_{1}( t) =\left\{ \begin{array}{ll@{ }l} 0 & & \hbox{if }t\lt 1 \\ 1 & & \hbox{if }t\geq 1 \end{array} \right. \) at \(c=1\)

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\(u_{3}( t) =\left\{ \begin{array}{ll@{ }l} 0 & & \hbox{if }t\lt 3 \\[3pt] 1 & & \hbox{if }t\geq 3 \end{array} \hbox{at }c=3\right. \)

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\({\lim\limits_{h\rightarrow 0}}\,{\dfrac{{( {x+h}) ^{2}-x^{2}}}{{h}}}\)

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\({\lim\limits_{h\rightarrow 0}}\,{\dfrac{{\sqrt{x+h}-\sqrt{x}}}{{h}}}\)

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\({\lim\limits_{h\rightarrow 0}}\,\dfrac{{{{ \dfrac{{1}}{{x+h}}}}-{{\dfrac{{1}}{{x}}}}}}{h}\)

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\({\lim\limits_{h\rightarrow 0}}\,{\dfrac{{{ \dfrac{{1}}{{( {x+h}) ^{3}}}}-{\dfrac{{1}}{{x^{3}}}}}}{{h}}}\)

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\({\lim\limits_{x\rightarrow 0}}\left[ {\dfrac{{1}}{{x}}}\left( \dfrac{{1}}{{4+x}}-\dfrac{1}{4}\right)\! \right]\)

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\(\lim\limits_{x\rightarrow -1}\left[ \dfrac{2}{x+1}\left( \dfrac{1}{3}-\dfrac{1}{x+4}\right) \!\right] \)

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\({\lim\limits_{x\rightarrow 7}}\,{\dfrac{{x-7}}{{\sqrt{x+2}-3}}}\)

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\({\lim\limits_{x\rightarrow 2}}\,\dfrac{x-2}{\sqrt{x+2}-2}\)

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\({\lim\limits_{x\rightarrow 1}}\,\dfrac{x^{3}-3x^{2}+3x-1}{x^{2}-2x+1}\)

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\({\lim\limits_{x\rightarrow -3}}\dfrac{x^{3}+7x^{2}+15x+9}{ x^{2}+6x+9}\)

92

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Cost of Water The Jericho Water District determines quarterly water costs, in dollars, using the following rate schedule:

Source: Jericho Water District, Syosset, NY.

1. Find a function \(C\) that models the quarterly cost, in dollars, of using \(x\) thousand gallons of water.
2. What is the domain of the function \(C?\)
3. Find each of the following limits. If the limit does not exist, explain why. \begin{equation*} \begin{array}{ll@{\quad}ll@{\quad}ll@{\quad}l} \lim\limits_{x\rightarrow 5}C( x) & & \lim\limits_{x\rightarrow 10}C( x) & & \lim\limits_{x\rightarrow 30}C( x) & & \lim\limits_{x\rightarrow 100}C( x) \end{array} \end{equation*}
4. What is \(\lim\limits_{x\rightarrow 0^{+}}C( x) ?\)
5. Graph the function \(C\).

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Cost of Natural Gas In February 2012 Peoples Energy had the following monthly rate schedule for natural gas usage in single- and two-family residences with a single gas meter:

Source: Peoples Energy, Chicago, IL.

1. Find a function \(C\) that models the monthly cost, in dollars, of using \(x\) therms of natural gas.
2. What is the domain of the function \(C?\)
3. Find \(\lim\limits_{x\rightarrow 50}C( x) \), if it exists. If the limit does not exist, explain why.
4. What is \(\lim\limits_{x\rightarrow 0^{+}}C( x) ?\)
5. Graph the function \(C.\)

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Low-Temperature Physics In thermodynamics, the average molecular kinetic energy (energy of motion) of a gas having molecules of mass \(m\) is directly proportional to its temperature \(T\) on the absolute (or Kelvin) scale. This can be expressed as \(\dfrac{1}{2}mv^{2}=\dfrac{3}{2}kT,\) where \(v=v(T)\) is the speed of a typical molecule at time \(t\) and \(k\) is a constant, known as the Boltzmann constant.

1. What limit does the molecular speed \(v\) approach as the gas temperature \(T\) approaches absolute zero (\(0{K}\) or \(-273^{\circ}{\rm C}\) or \(-469^{\circ}{\rm F}\))?
2. What does this limit suggest about the behavior of a gas as its temperature approaches absolute zero?

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For the function \(f(x) =\left\{ \begin{array}{c@{\qquad}rl} 3x+5 & \hbox{if} & x\leq 2 \\[3pt] 13-x & \hbox{if} & x>2 \end{array} \right. \), find

1. \({\lim\limits_{h\rightarrow 0^{-}}}{\dfrac{{f({2+h}) -f( {2}) }}{{h}}}\)
2. \({\lim\limits_{h\rightarrow 0^{+}}}\) \(\dfrac{{f( {2+h}) -f( {2}) }}{{h}}\)
3. Does \({\lim\limits_{h \rightarrow 0}}{\dfrac{{f( {2+h}) -f( {2}) }}{{h}}}\) exist?

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Use the fact that \(\vert x\vert =\left\{\! \begin{array}{l@{\quad}l} x & \hbox{if }x\geq 0 \\ -x & \hbox{if }x\lt 0 \end{array} \right. \) to show that \(\lim\limits_{x\rightarrow 0}\vert x\vert =0.\)

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Use the fact that \(\vert x\vert =\sqrt{x^{2}}\) to show that \(\lim\limits_{x\rightarrow 0}\vert x\vert =0.\)

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Find functions \(f\) and \(g\) for which \(\lim\limits_{x\rightarrow c}\) \([ f(x)+g(x)] \) may exist even though \(\lim\limits_{x\rightarrow c}f(x)\) and \(\lim\limits_{x\rightarrow c}g(x)\) do not exist.

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Find functions \(f\) and \(g\) for which \(\lim\limits_{x\rightarrow c}\) \([ f(x)g(x)] \) may exist even though \(\lim\limits_{x\rightarrow c}f(x)\) and \(\lim\limits_{x\rightarrow c}\,g( x) \) do not exist.

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Find functions \(f\) and \(g\) for which \(\lim\limits_{x\rightarrow c}\) \(\left[ \dfrac{f(x)}{g( x) }\right] \) may exist even though \(\lim\limits_{x\rightarrow c}f(x)\) and \(\lim\limits_{x \rightarrow c}g( x) \) do not exist.

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Find a function \(f\) for which \(\lim\limits_{x\rightarrow c}\vert f(x)\vert \) may exist even though \(\lim\limits_{x\rightarrow c}f(x)\) does not exist.

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Prove that if \(g\) is a function for which \(\lim\limits_{x\rightarrow c}g(x)\) exists and if \(k\) is any real number, then \(\lim\limits_{x\rightarrow c}[kg(x)]\) exists and \(\lim\limits_{x \rightarrow c}[kg(x)]=k\lim\limits_{x\rightarrow c}g(x)\).

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Prove that if the number \(c\) is in the domain of a rational function \(R( x) =\) \(\dfrac{p(x)}{q(x)}\), then \(\lim\limits_{x\rightarrow c}R( x) =R( c) \).

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Find \({\lim\limits_{x\rightarrow a}}{\dfrac{{x^{n}-a^{n}}}{{x-a}}}\), \(n\) a positive integer.

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Find \({\lim\limits_{x\rightarrow -a}}{\dfrac{{x^{n}+a^{n}}}{{x+a}}}\), \(n\) a positive integer.

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Find \({\lim\limits_{x\rightarrow 1}}\dfrac{x^{m}-1}{x^{n}-1}\), \(m\), \(n\) positive integers.

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Find \({\lim\limits_{x\rightarrow 0}}{\dfrac{{\sqrt[3]{{1+x}}-1}}{{x}}}\).

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Find \({\lim\limits_{x\rightarrow 0}}{\dfrac{{\sqrt{( {1+ax}) ( {1+bx}) }-1}}{{x}}}\).

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Find \({\lim\limits_{x\rightarrow 0}}{\dfrac{{\sqrt{ ( {1+a_{1}x}) ( {1+a_{2}x}) \cdots ( {1+a_{n}x}) }-1}}{{x}}}\).

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Find \({\lim\limits_{h\rightarrow 0}}{\dfrac{{f ({h}) -f (0)}}{{h}}}\) if \(f(x)=x|x|\).