Solution: find the limit of ƒ as (x,y)(0,0) or show that the

Problem 1E Find the limits in Exercise

Find the limits in Exercises 1–12. \(\lim _{(x, y) \rightarrow(0,4)} \frac{x}{\sqrt{y}}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (0,4) x/ sqrt y

Find the limits in Exercises 1–12. \(\lim _{(x, y) \rightarrow(3,4)} \sqrt{x^{2}+y^{2}-1}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (3,4) sqrt x^2 + y^2-1

Problem 4E Find the limits in Exercise

Problem 5E Find the limits in Exercise

Find the limits in Exercises 1–12. \(\lim _{(x, y) \rightarrow(2,-3)}\left(\frac{1}{x}+\frac{1}{y}\right)^{2}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (2,-3) (1/x+1/y)^2

Problem 8E Find the limits in Exercise

Find the limits in Exercises 1–12. \(\lim _{(x, y) \rightarrow(0, \ \ln 2)} e^{x-y}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (0, ln 2)e^x-y

Find the limits in Exercises 1–12. \(\lim _{(x, \ y) \rightarrow(0, 0)} \frac{e^{y} \sin x}{x}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (0,0)e^y sin x/x

Find the limits in Exercises 1–12. \(\lim _{(x, \ y) \rightarrow\left(1 / 27, \ \pi^{3}\right)} \cos \sqrt[3]{x y}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (1/27, pi^3) cos 3 sqrt xy

Problem 11E Find the limits in Exercise

Find the limits in Exercises 1–12. \(\lim _{(x, y) \rightarrow(\pi / 2,\ 0)} \frac{\cos y+1}{y-\sin x}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (pi/2, 0) cos y + 1/y-sin x

Find the limits in Exercises 13–24 by rewriting the fractions first. \(\lim _{(x, y) \rightarrow(1,1) \atop x \neq y} \frac{x^{2}-2 x y+y^{2}}{x - y}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (1, 1) x^2-2xy + y^2/x-y x not = y

Problem 14E Find the limits in Exercises 13–24 by rewriting the fractions first.

Find the limits in Exercises 13–24 by rewriting the fractions first. \(\lim _{(x, y) \rightarrow(1,1)} \frac{x y-y-2 x+2}{x-1}\) \(x \neq 1\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (1, 1)xy- y - 2x + 2/ x-1 x not = 1

Problem 16E Find the limits in Exercises 13–24 by rewriting the fractions first.

Find the limits in Exercises 13–24 by rewriting the fractions first. \(\lim _{x, y \rightarrow(0,0) \atop x \neq y} \frac{x-y+2 \sqrt{x}-2 \sqrt{y}}{\sqrt{x}-\sqrt{y}}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (0, 0)x - y + 2 sqrt x - 2 sqrt y/ sqrt x - sqrt y x not = y

Problem 18E Find the limits in Exercises 13–24 by rewriting the fractions first.

Find the limits in Exercises 13–24 by rewriting the fractions first. <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><munder><mrow><mfenced><mrow><mi mathvariant="normal">x</mi><mo>,</mo><mi mathvariant="normal">y</mi></mrow></mfenced><mo>&#x2192;</mo><mfenced><mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></mfenced></mrow><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mi mathvariant="normal">y</mi><mo>&#x2260;</mo><mn>4</mn></mrow></munder></munder><mfrac><mrow><msqrt><mn>2</mn><mi>x</mi><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mi>y</mi></msqrt><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>2</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mi>y</mi><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>4</mn></mrow></mfrac></math>

Problem 20E Find the limits in Exercises 13–24 by rewriting the fractions first.

Find the limits in Exercises 13–24 by rewriting the fractions first. \(\lim _{(x, y) \rightarrow(0,0)} \frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (0, 0) sin ( x^2 + y^2)/ x^2 + y^2

Find the limits in Exercises 13–24 by rewriting the fractions first. \(\lim _{(x, y) \rightarrow(2,0) \atop 2 x-y \neq 4} \frac{\sqrt{2 x-y}-2}{2 x-y-4}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (2, 0) sqrt 2x - y - 2 /2x - y - 4 2x - y not = 4

Problem 23E Find the limits in Exercises 13–24 by rewriting the fractions first.

Find the limits in Exercises 13–24 by rewriting the fractions first. \(\lim _{(x, y) \rightarrow(2,2)} \frac{x-y}{x^{4}-y^{4}}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (2, 2) x- y/x^4-y^4

Problem 25E Find the limits

Find the limits in Exercises 25–30. \(\lim _{P \rightarrow(1,-1,-1)} \frac{2 x y+y z}{x^{2}-z^{2}}\) Equation Transcription: Text Transcription: lim_ P right arrow (1, -1, -1) 2xy + yz/x^2-z^2

Problem 27E Find the limits

Problem 28E Find the limits

Find the limits in Exercises 25–30. \(\lim _{P \rightarrow(\pi, 0,3)} z e^{-2 y} \cos 2 x\) Equation Transcription: Text Transcription: lim_ P right arrow (pi, 0, 3) ze^-2y cos 2x

Find the limits in Exercises 25–30. \(\lim _{P \rightarrow(2,-3,6)} \ln \sqrt{x^{2}+y^{2}+z^{2}}\) Equation Transcription: Text Transcription: lim_ P right arrow (2, -3, 6) ln sqt x^2 + y^2 + z^2

Problem 31E At what points (x, y) in the plane are the functions in Exercises 31–34 continuous?

At what points \((x, y)\) in the plane are the functions in Exercises 31–34 continuous? a. \(f(x, y)=\frac{x+y}{x-y}\) b. \(f(x, y)=\frac{y}{x^{2}+1}\) Equation Transcription: Text Transcription: (x,y) f (x,y) =x + y/x - y f (x,y) =y/ x^2 + 1

At what points \((x, y)\) in the plane are the functions in Exercises 31–34 continuous? a. \(g(x, y)=\sin \frac{1}{x y}\) b. \(g(x, y)=\frac{x+y}{2+\cos x}\) Equation Transcription: Text Transcription: (x,y) g (x,y) =sin 1/xy g (x,y) =x + y/2 + cos x

Problem 34E At what points (x, y) in the plane are the functions in Exercises 31–34 continuous?

At what points \((x, y, z)\) in space are the functions in Exercises 35–40 continuous? a. \(f(x, y, z)=x^{2}+y^{2}-2 z^{2}\) b. \(f(x, y, z)=\sqrt{x^{2}+y^{2}-1}\) Equation Transcription: Text Transcription: (x,y, z) f (x, y, z) =x^2+ y^2 - 2z^2 f (x, y, z) = sqrt x^2 + y^2-1

Problem 36E At what points (x, y, z) in space are the functions in Exercises 35–40 continuous?

At what points \((x, y, z)\) in space are the functions in Exercises 35–40 continuous? a. \(h(x, y, z)=x y \sin \frac{1}{z}\) b. \(h(x, y, z)=\frac{1}{x^{2}+z^{2}-1}\) Equation Transcription: Text Transcription: (x,y, z) h(x, y, z) = xy sin 1z h(x, y, z) = 1x2 + z2 - 1

Problem 38E At what points (x, y, z) in space are the functions in Exercises 35–40 continuous?

At what points (x, y, z) in space are the functions in Exercises 35–40 continuous? a. b.

Problem 40E At what points (x, y, z) in space are the functions in Exercises 35–40 continuous?

By considering different paths of approach, show that the functions in Exercises 41–48 have no limit as \((x, y) \rightarrow(0.0)\). \(f(x, y)=-\frac{x}{\sqrt{x^{2}+y^{2}}}\) Equation Transcription: Text Transcription: (x, y) right arrow (0. 0) f(x, y) = -x/ sqrt x^2 + y^2

By considering different paths of approach, show that the functions in Exercises 41–48 have no limit as \((x, y) \rightarrow(0.0)\). \(f(x, y)=\frac{x^{4}}{x^{4}+y^{2}}\) Equation Transcription: Text Transcription: (x,y, z) f(x, y) = x^4/x^4 + y^2

Problem 43E By considering different paths of approach, show that the functions in Exercises 41–48 have no limit as (x,y) ?(0,0).

By considering different paths of approach, show that the functions in Exercises 41–48 have no limit as \((x, y) \rightarrow(0.0)\). \(f(x, y)=\frac{x y}{|x y|}\) Equation Transcription: Text Transcription: (x,y, z) f(x, y) = xy/ |xy|

By considering different paths of approach, show that the functions in Exercises 41–48 have no limit as \((x, y) \rightarrow(0.0)\). \(g(x, y)=\frac{x-y}{x+y}\) Equation Transcription: Text Transcription: (x,y, z) g(x, y) = x - y/x + y

Problem 46E By considering different paths of approach, show that the functions in Exercises 41–48 have no limit as (x,y) ?(0,0).

Problem 47E By considering different paths of approach, show that the functions in Exercises 41–48 have no limit as (x,y) ?(0,0).

In Exercises 49 and 50, show that the limits do not exist. \(\lim _{(x, y) \rightarrow(1,1)} \frac{x y^{2}-1}{y-1}\) Equation Transcription: Text Transcription: lim_ (x,y) right arrow (1, 1) xy^2 - 1/ y-1

By considering different paths of approach, show that the functions in Exercises 41–48 have no limit as \((x, y) \rightarrow(0,0)\). \(h(x, y)=\frac{x^{2} y}{x^{4}+y^{2}}\) Equation Transcription: Text Transcription: (x,y) rightarrrow (0,0) h(x, y) = x^2y/x^4 + y^2

Problem 50E In Exercises 49 and 50, show that the limits do not exist.

Let \(f(x, y)=\left\{\begin{array}{ll} 1, & y \geq x^{4} \\ 1, & y \leq 0 \\ 0, & otherwise \end{array}\right. \) Find each of the following limits, or explain that the limit does not exist. a. \(\lim _{(x, y) \rightarrow(0, 1)} f(x, y)\) b. \(\lim _{(x, y) \rightarrow(2, 3)} f(x, y)\) c. \(\lim _{(x, y) \rightarrow(0, 0)} f(x, y)\) Equation Transcription: { Text Transcription: f(x,y)={1, y geq 0 1 y leq 0 0, otheriwise lim_(x,y) rightarrow (0,1) f(x,y) lim_(x,y) rightarrow (2,3) f(x,y) lim_(x,y) rightarrow (0,0) f(x,y)

Let \(f(x, y)=\left\{\begin{array}{ll} x^{2}, & x \geq 2 \\ x^{3}, & x<2 \end{array}\right. \) Find the following limits. a. \(\lim _{(x, y) \rightarrow(3,-2)} f(x, y)\) b. \(\lim _{(x, y) \rightarrow(-2,1)} f(x, y)\) c. \(\lim _{(x, y) \rightarrow(0, 0)} f(x, y)\) Equation Transcription: { Text Transcription: f(x)={x^3, x<2 x^2, x geq 2 lim_(x,y) rightarrow (3,-2) f(x,y) lim_(x,y) rightarrow (-2,1) f(x,y) lim_(x,y) rightarrow (0,0) f(x,y)

Problem 53E Show that the function in Example 6 has limit 0 along every straight line approaching (0, 0).

Problem 54E If f(x0,y0)=3, what can you say about if ƒ is continuous at If ƒ is not continuous at Give reasons for your answers.

Does knowing that \(1-\frac{x^{2} y^{2}}{3}<\frac{\tan ^{-1} x y}{x y}<1\) tell you anything about \(\lim _{(x, y) \rightarrow(0,0)} \frac{\tan ^{-1} x y}{x y}\) Give reasons for your answer. Equation Transcription: Text Transcription: 1-x^2y^2/3 < tan^-1 xy/xy < 1 lim_(x,y) rightarrow (0,0) tan^-1 xy/xy

Does knowing that \(2|x y|-\frac{x^{2} y^{2}}{6}<4-4 \cos \sqrt{|x y|}<2|x y|\) tell you anything about \(\lim _{(x, y) \rightarrow(0,0)} \frac{4-4 \cos \sqrt{|x y|}}{|x y|}\) Give reasons for your answer. Equation Transcription: Text Transcription: 2|xy|-x^2y^2/6 < 4-4 cos sqrt |xy| < 2|xy| lim_(x,y) rightarrow (0,0) 4-4 cos sqrt |xy|/|xy|

Problem 57E Does knowing that tell you anything about Give reasons for your answer.

Does knowing that \(|\cos (1 / y)| \leq 1\) tell you anything about \(\lim _{(x, y) \rightarrow(0,0)} x \cos \frac{1}{y}\) Give reasons for your answer. Equation Transcription: 1 Text Transcription: |cos (1/y)| leq 1 lim_(x,y) rightarrow (0,0) x cos 1/y

Problem 59E (Continuation of Example 5.) a. Reread Example 5. Then substitute into the formula and simplify the result to show how the value of ƒ varies with the line’s angle of inclination. b. Use the formula you obtained in part (a) to show that the limit of ƒ as along the line varies from -1 to 1 depending on the angle of approach.

Continuous extension Define in a way that extends \(f(x, y)=x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}}\) to be continuous at the origin. Equation Transcription: Text Transcription: ƒ(0, 0) f(x,y)=xy x^2-y^2/x^2+y^2

In Exercises 61–66, find the limit of ƒ as \((x, y) \rightarrow(0,0)\) or show that the limit does not exist. \(f(x, y)=\frac{x^{3}-x y^{2}}{x^{2}+y^{2}}\) Equation Transcription: Text Transcription: (x,y) rightarrow (0,0) f(x, y) = x^3 - xy^2/x^2 + y^2

Problem 62E find the limit of ƒ as (x,y)?(0,0) or show that the limit does not exist.

In Exercises 61–66, find the limit of ƒ as \((x, y) \rightarrow(0,0)\) or show that the limit does not exist. \(f(x, y)=\frac{y^{2}}{x^{2}+y^{2}}\) Equation Transcription: Text Transcription: (x,y) rightarrow (0,0) ƒ(x, y) = y^2/x^2 + y^2

Problem 64E find the limit of ƒ as (x,y)?(0,0) or show that the limit does not exist.

In Exercises 61–66, find the limit of ƒ as \((x, y) \rightarrow(0,0)\) or show that the limit does not exist. \(f(x, y)=\tan ^{-1}\left(\frac{|x|+|y|}{x^{2}+y^{2}}\right)\) Equation Transcription: Text Transcription: (x,y) rightarrow (0,0) ƒ(x, y) = tan^-1 (| x | + | y |/x^2 + y^2)

Problem 66E find the limit of ƒ as (x,y)?(0,0) or show that the limit does not exist.

In Exercises 67 and 68, define ƒ(0, 0) in a way that extends ƒ to be continuous at the origin. \(f(x, y)=\ln \left(\frac{3 x^{2}-x^{2} y^{2}+3 y^{2}}{x^{2}+y^{2}}\right)\) Equation Transcription: Text Transcription: ƒ(x, y) = ln (3x^2-x^2y^2 + 3y^2/x^2 + y^2)

Problem 68E define ƒ(0, 0) in a way that extends ƒ to be continuous at the origin.

Each of Exercises 69–74 gives a function \(f(x, y)\) and a positive number \(\epsilon\). In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y)\), \(f(x, y)=x^{2}+y^{2}, \quad \epsilon=0.01\) Equation Transcription: Text Transcription: ƒ(x, y) delta ? 0 (x, y) epsilon f(x, y) = x^2 + y^2, epsilon = 0.01

Each of Exercises 69–74 gives a function \(f(x, y)\) and a positive number \(\epsilon\). In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y)\), \(f(x, y)=y /\left(x^{2}+1\right), \quad \epsilon=0.05\) Equation Transcription: Text Transcription: ƒ(x, y) delta ? 0 (x, y) epsilon f(x, y) = y/(x^2+1), epsilon=0.05

Each of Exercises 69–74 gives a function \(f(x, y)\) and a positive number \(\epsilon\). In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y)\), \(f(x, y)=(x+y) /\left(x^{2}+1\right), \quad \epsilon=0.01\) Equation Transcription: Text Transcription: ƒ(x, y) delta ? 0 (x, y) epsilon f(x, y) = (x+y)/(x^2+1), epsilon=0.01

Problem 72E Each of Exercises 69–74 gives a function ƒ(x, y) and a positive number In each exercise, show that there exists a ? > 0 such that for all (x, y),

Problem 73E Each of Exercises 69–74 gives a function ƒ(x, y) and a positive number In each exercise, show that there exists a ? > 0 such that for all (x, y),

Problem 74E Each of Exercises 69–74 gives a function ƒ(x, y) and a positive number In each exercise, show that there exists a ? > 0 such that for all (x, y),

Each of Exercises 75–78 gives a function \(f(\mathrm{x}, \mathrm{y}, \mathrm{z})\) and a positive number \(\epsilon\). In each exercise, show that there exists a \(\delta>0\) such that for all \((\mathrm{x}, \mathrm{y}, \mathrm{z})\), \(\sqrt{x^{2}+y^{2}+z^{2}}<\delta \Rightarrow|f(x, y, z)-f(0,0,0)|<\epsilon\). \(f(x, y, z)=x^{2}+y^{2}+z^{2}, \quad \epsilon=0.015\) Equation Transcription: ƒ(x, y, z) (x, y, z) Text Transcription: ƒ(x, y, z) epsilon delta ? 0 (x, y, z) sqrt x^2+y^2+z^2 < delta right double arrow |f(x,y,z)-f(0, 0, 0)| < epsilon f(x, y, z) = x^2 + y^2 + z^2, epsilon= 0.015

Each of Exercises 75–78 gives a function \(f(\mathrm{x}, \mathrm{y}, \mathrm{z})\) and a positive number \(\epsilon\). In each exercise, show that there exists a \(\delta>0\) such that for all \((\mathrm{x}, \mathrm{y}, \mathrm{z})\), \(\sqrt{x^{2}+y^{2}+z^{2}}<\delta \Rightarrow|f(x, y, z)-f(0,0,0)|<\epsilon\). \(f(x, y, z)=x y z, \quad \epsilon=0.008\) Equation Transcription: ƒ(x, y, z) (x, y, z) Text Transcription: ƒ(x, y, z) epsilon delta ? 0 (x, y, z) sqrt x^2+y^2+z^2 < delta right double arrow |f(x,y,z)-f(0, 0, 0)| < epsilon f(x, y, z) = xyz, epsilon = 0.008

Problem 77E Each of Exercises 75–78 gives a function ƒ(x, y) and a positive number In each exercise, show that there exists a ? > 0 such that for all (x, y),

Each of Exercises 75–78 gives a function \(f(\mathrm{x}, \mathrm{y}, \mathrm{z})\) and a positive number \(\epsilon\). In each exercise, show that there exists a \(\delta>0\) such that for all \((\mathrm{x}, \mathrm{y}, \mathrm{z})\), \(\sqrt{x^{2}+y^{2}+z^{2}}<\delta \Rightarrow|f(x, y, z)-f(0,0,0)|<\epsilon\). \(f(x, y, z)=\tan ^{2} x+\tan ^{2} y+\tan ^{2} z, \quad \epsilon=0.03\) Equation Transcription: ƒ(x, y, z) (x, y, z) Text Transcription: ƒ(x, y, z) epsilon delta ? 0 (x, y, z) sqrt x^2+y^2+z^2 < delta right double arrow |f(x,y,z)-f(0, 0, 0)| < epsilon f(x, y, z) = tan^2 x + tan^2 y + tan^2 z, epsilon = 0.03

Show that \(f(x, y, z)=x+y-z\) is continuous at every point \(\left(x_{0}, y_{0}, z_{0}\right)\). Equation Transcription: Text Transcription: f(x, y, z) = x + y - z (x_0, y_0, z_0).

Problem 80E Show that is continuous at the origin.