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Multiple Choice  The set of all points \(P\) for which the distance \(d(P, P_{0})<\delta \) is called [(a) a \(\delta\) position, (b) an \(\varepsilon \) -value, (c) a \(\delta \)-neighborhood] of \(P_{0}\).

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\(\lim\limits_{(x, y)\rightarrow (1, 8)}( 2y) =\)__________.

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True or False  One way to show that \( \lim\limits_{(x, y)\rightarrow (0, 0)}f(x)=L\) is to find two lines \( y=m_{1}x\) and \(y=m_{2}x\) for which \(\lim\limits_{(x, y)\rightarrow (0, 0)}f(x)=L.\)

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True or False  In order for the limit of a function \(f\) to exist at a point \(P_{0}\), the limit of \(f\) must be the same along every curve in the domain of \(f\) that contains \(P_{0}\).

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Multiple Choice  A point \(P_{0}\) is [(a) an interior, (b) an exterior, (c) a boundary, (d) an isolated] point of \(S\) if every \(\delta \)-neighborhood of \(P_{0}\) contains points both in \( S\) and not in \(S\).

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Multiple Choice  If every point of \(S\) is an interior point, then \(S\) is [(a) an open, (b) a closed, (c) an interior] set.

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True or False  If two functions \(f\) and \(g\) are both continuous at the point \(P_{0},\) then the quotient \(\dfrac{g}{f}\) is also continuous at \(P_{0}.\)

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\(\lim\limits_{(x,y) \rightarrow ( 0,0) }e^{x^{2}+y^{2}}=\)__________.

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\(\lim\limits_{(x, y)\rightarrow (1,2)}(x^{2}+xy-y^{2}+8)\)

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

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

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\(\lim\limits_{(x, y,z)\rightarrow (0,1,-1)}(x^{2}-y^{2}z^{2})\)

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\(\lim\limits_{(x, y)\rightarrow ( {\pi }/{2},\pi ) }(\sin x\cos y)\)

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\(\lim\limits_{(x, y)\rightarrow (2,e)}( x^{2}y\ln y) \)

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\(\lim\limits_{(x, y)\rightarrow (1, 5)}\dfrac{4x-xy+4}{4y-y^{2}}\)

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

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\(\lim\limits_{(x, y)\rightarrow (\pi , 0)}\dfrac{\cos x\cos y}{x}\)

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\(\lim\limits_{(x, y)\rightarrow (\pi ,\pi )}\dfrac{\cos y(1-\cos x)}{xy}\)

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\(\lim\limits_{(x, y)\rightarrow (0, 0)}\dfrac{e^{x}-4}{e^{y}}\)

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\(\lim\limits_{(x, y)\rightarrow (0,\, 0)}\dfrac{e^{x}\cos y-\cos y}{e^{y}}\)

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\(\lim\limits_{(x, y)\rightarrow (2, 1)}\dfrac{x^{2}+xy-6y^{2}}{x^{2}+4y^{2}}\)

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\(\lim\limits_{(x, y)\rightarrow (0,-2)}\dfrac{y^{2}+xy+4y+e^{x}}{xy-y+2x-e^{x}}\)

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\(\lim\limits_{(x, y)\rightarrow (2, 0)}\dfrac{x^{2}y+x}{x^{3}y+3xy^{2}-8}\)

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

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\(\lim\limits_{(x, y)\rightarrow (0, 0)}\dfrac{3xy}{2x^{2}+y^{2}}\)

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\(\lim\limits_{(x, y)\rightarrow (0, 0)} \dfrac{2xy}{x^{2}+3y^{2}}\)

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\(\lim\limits_{(x, y) \rightarrow (0, 0)}\dfrac{xy^{2}}{x^{2}+y^{3}}\)

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\(\lim\limits_{(x, y)\rightarrow (0, 0)}\dfrac{2x^{2}y}{3x^{3}+y^{2}}\)

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

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

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\(\lim\limits_{(x, y)\rightarrow (0, 0)}\dfrac{x^{2}+xy}{x^{2}+y^{2}}\)

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

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

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\(\lim\limits_{(x, y) \rightarrow (0, 0)}\dfrac{2xy}{x^{2}+y^{2}} \)

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

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\(\lim\limits_{(x, y)\rightarrow (0, 0)}\dfrac{x^{2}+y^{4}}{x^{2}+y^{2}}\)

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\(f(x,y) =3x^{2}y-4x^{2}y^{2}+10xy^{2}-9 \)

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\(f(x,y) =x^{3}+2x^{2}y+xy^{2}-4y^{3}\)

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\(f(x,y) =\dfrac{x^{2}-y^{2}}{x-y}\)

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\(f(x,y) =\dfrac{2x^{2}y+xy^{2}}{1-xy}\)

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\(f(x,y) =e^{x^{2}-y^{2}}\)

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\(f(x,y) =\ln ( x^{2}+y^{2}) \)

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\(f(x,y) =\sin (x^{2}-y)\)

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\(f(x,y) =\cos \sqrt{x^{2}-y}\)

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\(f(x,y) =\sin (x+y)\cos (x-y)\)

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\(f(x,y) =e^{x}\sin (xy)\)

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\(f(x,y) =\dfrac{x+3xy^{2}}{e^{x^{2}-y^{2}}}\)

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\(f(x,y) =\dfrac{x+3xy^{2}}{\ln (x^{2}+y^{2}) }\)

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\(\lim\limits_{(x, y)\rightarrow (1, 0)}\dfrac{x^{2}-y^{2}}{x-y}\)

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

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\(\lim\limits_{(x, y)\rightarrow (1, 1)}e^{x^{2}-y^{2}}\)

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\(\lim\limits_{(x, y) \rightarrow (0, e)}\ln (x^{2}+y^{2})\)

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\(\lim\limits_{(x, y)\rightarrow ( {\pi }/{ 2}, \pi ) }[ \sin (x+y)\cos (x-y)] \)

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\(\lim\limits_{(x, y)\rightarrow (0, \pi /{2})} e^{x}\sin (xy)\)

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\(\lim\limits_{(x, y)\rightarrow (\pi , 0)}\dfrac{e^{x^{2}+y^{2}}\cos x^{2}}{\cos y^{2}}\)

829

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\(\lim\limits_{(x, y)\rightarrow ( {\pi }/{2}, 4) }\dfrac{e^{x^{2}y}}{\cos ( 2x) }\)

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\(\lim\limits_{(x, y)\rightarrow (0, 0)} \tan ^{-1}\left( \dfrac{e^{x+y}}{y^{2}+1}\right) \)

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\(\lim\limits_{(x, y)\rightarrow (\pi, 0)} \tan ^{-1}[ \cos ( x+y) ] \)

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the line \(x=t\), \(y=t\), \(z=t\)

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the line \(x=2t\), \(y=3t\), \(z=4t\)

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the curve \(x=t\), \(y=t^{2}\), \(z=t^{2}\)

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the line \(x=at\), \(y=bt\), \(z=ct\), \(b^{2}+c^{2}>0\)

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\(\lim\limits_{(x, y)\rightarrow (0,\, 0)}\dfrac{x^{2}y}{x^{2}+y^{2}}=0\)

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\(\lim\limits_{(x, y)\rightarrow (0,\, 0)}\dfrac{\sin ( x^{2}+y^{2})}{x^2+y^2}=1\)

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\(f(x, y)=\dfrac{xy^{2}}{x^{2}+y^{2}}\)

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\(f(x, y)=\dfrac{x^{2}y}{x^{2}+y^{2}}\)

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\(f(x, y)=\dfrac{2x^{2}+y^{2}}{x^{2}+y^{2}}\)

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\(f(x, y)=\dfrac{x^{4}-y^{2}}{x^{2}+y^{2}}\)

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\(f(x,y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} \dfrac{3xy}{x^{2}+y^{2}}~ & \hbox{if } & ~(x,y)\neq (0,0) \\[3pt] 0 & \hbox{if} & (x,y)=(0,0) \end{array} \right. \)

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\(f(x,y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} \dfrac{\sin ( xy) }{x^{2}+y^{2}} & \hbox{if } & (x,y)\neq (0,0) \\[3pt] 1 & \hbox{if} & (x,y)=(0,0) \end{array} \right.\)

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\(f(x, y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} \dfrac{\sin (x^{2}+y^{2})}{x^{2}+y^{2}} & \hbox{if} & (x, y)\neq (0, 0) \\[3pt] 1 & \hbox{if} & (x, y)=(0, 0) \end{array} \right. \)

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\(f(x, y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} \dfrac{\sin (x^{2}-y^{2})}{x^{2}+y^{2}} & \hbox{if} & (x, y)\neq (0, 0) \\[3pt] 1 & \hbox{if} & (x, y)=(0, 0) \end{array} \right. \)

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

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\(\lim\limits_{(x, y)\rightarrow (0, 0)}\dfrac{xy^{2}}{x^{2}+y^{2}}\)

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\(\lim\limits_{(x, y)\rightarrow (0, 0)}\dfrac{\cos (x^{2}+y^{2})}{x^{2}+y^{2}}\)

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\(\lim\limits_{(x, y)\rightarrow (0, 0)}\dfrac{\sin (x^{2}+y^{2})}{x^{2}+y^{2}}\)

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Find \(\lim\limits_{(x,y) \rightarrow (0,0) }f(x,y) ,\) if \(f(x,y) =\dfrac{ x^{3}-4x^{2}y+4xy^{2}+5x-10y}{x-2y}.\)