Processing math: 3%

12.2 Assess Your Understanding

Printed Page 828

828

Concepts and Vocabulary

  1. Multiple Choice  The set of all points P for which the distance d(P,P0)<δ is called [(a) a δ position, (b) an ε -value, (c) a δ-neighborhood] of P0.

(c)

  1. lim__________.

16

  1. 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.

False

  1. 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}.

True

  1. 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.

{\bf (c)}

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

{\bf (a)}

  1. 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}.

False

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

1

Skill Building

In Problems 9–24, use algebraic properties of limits of two variables (p. 821) to find each limit.

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

7

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

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

-1

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

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

-1

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

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

-\dfrac35

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

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

-\dfrac1{\pi}

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

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

-3

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

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

0

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

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

-\dfrac{1}4

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

In Problems 25–32, find each limit by approaching (0, 0) along:

  1. (a) The x-axis.
  2. (b) The y-axis.
  3. (c) The line y=x.
  4. (d) The line y=3x.
  5. (e) The parabola y=x^{2}.

Can you conclude anything?

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

  1. (a) 0
  2. (b) 0
  3. (c) 1
  4. (d) \dfrac9{11}
  5. (e) 0. This limit does not exist.

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

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

  1. (a) 0
  2. (b) 0
  3. (c) 0
  4. (d) 0
  5. (e) 0. No conclusion can be reached.

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

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

  1. (a) 0
  2. (b) 0
  3. (c) \dfrac32
  4. (d) \dfrac{27}{82}
  5. (e) 0. This limit does not exist.

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

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

  1. (a) 1
  2. (b) 0
  3. (c) 1
  4. (d) \dfrac25
  5. (e) 1. This limit does not exist.

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

In Problems 33–36, show that the limit does not exist.

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

See Student Solutions Manual.

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

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

See Student Solutions Manual.

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

In Problems 37–48, determine where each function is continuous.

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

This function is continuous for all points (x,y) in the plane.

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

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

This function is continuous for all points (x,y) in the plane except on the line y=x.

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

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

This function is continuous for all points (x,y) in the plane.

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

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

This function is continuous for all points (x,y) in the plane.

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

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

This function is continuous for all points (x,y) in the plane.

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

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

This function is continuous for all points (x,y) in the plane.

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

In Problems 49–58, find each limit.

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

1

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

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

1

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

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

0

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

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

e^{\pi^2}\cos \pi^2

829

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

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

\dfrac{\pi}4

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

Applications and Extensions

In Problems 59–62, find \lim\limits_{(x, y, z)\rightarrow (0, 0, 0)}\dfrac{2yz}{x^{4}+y^{2}+z^{2}} along the indicated curves.

  1. the line x=t, y=t, z=t

1

  1. the line x=2t, y=3t, z=4t

  1. the curve x=t, y=t^{2}, z=t^{2}

\dfrac23

  1. the line x=at, y=bt, z=ct, b^{2}+c^{2}>0

In Problems 63 and 64, use the \varepsilon-\delta definition of a limit to prove each limit statement.

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

See Students Solutions Manual.

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

In Problems 65–72,

  1. (a) Determine whether each function f is continuous at {\rm (0, 0)}.
  2. (b) If f is discontinuous at {\rm (0, 0)}, is it possible to define f{\rm (0, 0)} so that f would be continuous at {\rm (0, 0)}?
  3. (c) If the answer to (b) is yes, how should f{\rm (0, 0)} be defined?

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

  1. (a) The function is discontinuous at (0,0).
  2. (b) Yes, it is possible to define f(x,y) at (0,0).
  3. (c) 0

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

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

  1. (a) The function is discontinuous at (0,0).
  2. (b) No, it is not possible to define f(x,y) at (0,0).

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

  1. 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.

  1. (a) The function is discontinuous at (0,0).
  2. (b) No, it is not possible to define f(x,y) at (0,0).

  1. 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.

  1. 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.

  1. (a) The function is continuous at (0,0).

  1. 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.

Challenge Problems

In Problems 73–76, find each limit by converting to polar coordinates.

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

0

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

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

This limit does not exist.

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

  1. 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}.

5