In Exercises 85 and 86, use spherical coordinates to find the limit. [Hint: Let and and

In Exercises 1–4, use the definition of the limit of a function of two variables to verify the limit. \(\lim _{(x, y) \rightarrow(1,0)} x=1\) Text Transcription: lim_(x, y) rightarrow (1,0) x=1

In Exercises 1–4, use the definition of the limit of a function of two variables to verify the limit. \(\lim _{(x, y) \rightarrow(4,-1)} x=4\) Text Transcription: lim_(x, y) rightarrow (4,-1) x=4

In Exercises 1–4, use the definition of the limit of a function of two variables to verify the limit. \(\lim _{(x, y) \rightarrow(1,-3)} y=-3\) Text Transcription: lim_(x, y) rightarrow (1,-3) y=-3

In Exercises 1–4, use the definition of the limit of a function of two variables to verify the limit. \(\lim _{(x, y) \rightarrow(a, b)} y=b\) Text Transcription: lim_(x, y) rightarrow(a, b) y=b

In Exercises 5–8, find the indicated limit by using the limits \(\lim _{(x, y) \rightarrow(a, b)} f(x, y)=4\) and \(\lim _{(x, y) \rightarrow(a, b)} g(x, y)=3\). \(\lim _{(x, y) \rightarrow(a, b)}[f(x, y)-g(x, y)]\) Text Transcription: lim_(x, y) rightarrow(a, b) f(x, y) = 4 lim_(x, y) rightarrow(a, b) g(x, y) = 3 lim_(x, y) rightarrow (a, b) [f(x, y) - g(x, y)]

In Exercises 5–8, find the indicated limit by using the limits \(\lim _{(x, y) \rightarrow(a, b)} f(x, y)=4\) and \(\lim _{(x, y) \rightarrow(a, b)} g(x, y)=3\). \(\lim _{(x, y) \rightarrow(u, b)}\left[\frac{5 f(x, y)}{g(x, y)}\right]\) Text Transcription: lim_(x, y) rightarrow (a, b) f(x, y) = 4 lim_(x, y) rightarrow (a, b) g(x, y) = 3 lim_(x, y) rightarrow (u, b) [5 f(x, y) / g(x, y)]

In Exercises 5–8, find the indicated limit by using the limits \(\lim _{(x, y) \rightarrow(a, b)} f(x, y)=4\) and \(\lim _{(x, y) \rightarrow(a, b)} g(x, y)=3\). \(\lim _{(x, y) \rightarrow(a, b)}[f(x, y) \ g(x, y)]\) Text Transcription: lim_(x, y) rightarrow (a, b) f(x, y) = 4 lim_(x, y) rightarrow (a, b) g(x, y) = 3 lim_(x, y) rightarrow (a, b) [f(x, y) g(x, y)]

In Exercises 5–8, find the indicated limit by using the limits \(\lim _{(x, y) \rightarrow(a, b)} f(x, y)=4\) and \(\lim _{(x, y) \rightarrow(a, b)} g(x, y)=3\). \(\lim _{(x, y) \rightarrow(a, b)}\left[\frac{f(x, y)+g(x, y)}{f(x, y)}\right]\) Text Transcription: lim_(x, y) rightarrow (a, b) f(x, y) = 4 lim_(x, y) rightarrow (a, b) g(x, y) = 3 lim_(x, y) rightarrow (a, b) [f(x, y)+g(x, y) / f(x, y)]

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(2,1)}\left(2 x^{2}+y\right)\) Text Transcription: lim_(x, y) rightarrow (2,1) (2x^2 + y)

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(0,0)}(x+4 y+1)\) Text Transcription: lim_(x, y) rightarrow (0,0) (x+4y+1)

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(1,2)} e^{x y}\) Text Transcription: lim_(x, y) rightarrow (1,2) e^xy

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(2,4)} \frac{x+y}{x^{2}+1}\) Text Transcription: lim_(x, y) rightarrow (2,4) x+y / x^2 + 1

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(0,2)} \frac{x}{y}\) Text Transcription: lim_(x, y) rightarrow (0,2) x/y

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(-1,2)} \frac{x+y}{x-y}\) Text Transcription: lim_(x, y) rightarrow (-1,2) x+y / x-y

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(1,1)} \frac{x y}{x^{2}+y^{2}}\) Text Transcription: lim_(x, y) rightarrow (1,1) xy / x^2 + y^2

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(1,1)} \frac{x}{\sqrt{x+y}}\) Text Transcription: lim_(x, y) rightarrow (1,1) x / sqrt x+y

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(\pi / 4,2)} y \cos x y\) Text Transcription: lim_(x, y) rightarrow (pi/4,2) y cos xy

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(2 \pi, 4)} \sin \frac{x}{y}\) Text Transcription: lim_(x, y) rightarrow (2pi, 4) sin x/y

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(0,1)} \frac{\arcsin x y}{1-x y}\) Text Transcription: lim_(x, y) rightarrow (0,1) arcsin xy / 1-xy

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y) \rightarrow(0,1)} \frac{\arccos (x / y)}{1+x y}\) Text Transcription: lim_(x, y) rightarrow (0,1) arccos (x/y) / 1+xy

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y, z) \rightarrow(1,3,4)} \sqrt{x+y+z}\) Text Transcription: lim_(x, y, z) rightarrow (1,3,4) sqrt x+y+z

In Exercises 9–22, find the limit and discuss the continuity of the function. \(\lim _{(x, y, z) \rightarrow(-2,1,0)} x e^{y z}\) Text Transcription: lim_(x, y, z) rightarrow (-2,1,0) xe^yz

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(1,1)} \frac{x y-1}{1+x y}\) Text Transcription: lim_(x, y) rightarrow (1,1) xy-1 / 1+xy

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(1,-1)} \frac{x^{2} y}{1+x y^{2}}\) Text Transcription: lim_(x, y) rightarrow (1,-1) x^2 y / 1+xy^2

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(0,0)} \frac{1}{x+y}\) Text Transcription: lim_(x, y) rightarrow (0,0) 1 / x+y

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(0,0)} \frac{1}{x^{2} y^{2}}\) Text Transcription: lim_(x, y) rightarrow (0,0) 1 / x^2 y^2

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(2,2)} \frac{x^{2}-y^{2}}{x-y}\) Text Transcription: lim_(x, y) rightarrow (2,2) x^2 - y^2 / x-y

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-4 y^{4}}{x^{2}+2 y^{2}}\) Text Transcription: lim_(x, y) rightarrow (0,0) x^4 - 4y^4 / x^2 + 2y^2

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x-y}{\sqrt{x}-\sqrt{y}}\) Text Transcription: lim_(x, y) rightarrow (0,0) x-y / sqrt x - sqrt y

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(2,1)} \frac{x-y-1}{\sqrt{x-y}-1}\) Text Transcription: lim_(x, y) rightarrow (2,1) x-y-1 / sqrt x-y - 1

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x+y}{x^{2}+y}\) Text Transcription: lim_(x, y) rightarrow (0,0) x+y / x^2 + y

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x}{x^{2}-y^{2}}\) Text Transcription: lim_(x, y) rightarrow (0,0) x / x^2 - y^2

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}}{\left(x^{2}+1\right)\left(y^{2}+1\right)}\) Text Transcription: lim_(x, y) rightarrow(0,0) x^{2} / (x^2 + 1) (y^2 + 1)

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y) \rightarrow(0,0)} \ln \left(x^{2}+y^{2}\right)\) Text Transcription: lim_(x, y) rightarrow (0,0) ln (x^2 + y^2)

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x y+y z+x z}{x^{2}+y^{2}+z^{2}}\) Text Transcription: lim_(x, y, z) rightarrow (0,0,0) xy+yz+xz / x^2 + y^2 + z^2

In Exercises 23–36, find the limit (if it exists). If the limit does not exist, explain why. \(\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x y+y z^{2}+x z^{2}}{x^{2}+y^{2}+z^{2}}\) Text Transcription: lim_(x, y, z) rightarrow (0,0,0) xy+yz^2 + xz^2 / x^2 + y^2 + z^2

In Exercises 37 and 38, discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as \((x, y) \rightarrow(0,0)\) \(f(x, y)=e^{x y}\) Text Transcription: (x, y) rightarrow (0,0) f(x, y)=e^xy

In Exercises 37 and 38, discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as \((x, y) \rightarrow(0,0)\) \(f(x, y)=1-\frac{\cos \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}\) Text Transcription: (x, y) rightarrow (0,0) f(x, y)=1-cos (x^2 + y^2) / x^2 + y^2

In Exercises 39–42, use a graphing utility to make a table showing the values of f(x, y) at the given points for each path. Use the result to make a conjecture about the limit of f(x, y) as \((x, y) \rightarrow(0,0)\). Determine whether the limit exists analytically and discuss the continuity of the function. \(f(x, y)=\frac{x y}{x^{2}+y^{2}}\) Path: y = 0 Points: (1, 0), (0.5, 0), (0.1, 0), (0.01, 0), (0.001, 0) Path: y = x Points: (1, 1), (0.5, 0.5), (0.1, 0.1), (0.01, 0.01), (0.001, 0.001) Text Transcription: (x, y) rightarrow (0,0) f(x, y)=xy / x^2 + y^2

In Exercises 39–42, use a graphing utility to make a table showing the values of f(x, y) at the given points for each path. Use the result to make a conjecture about the limit of f(x, y) as \((x, y) \rightarrow(0,0)\). Determine whether the limit exists analytically and discuss the continuity of the function. \(f(x, y)=\frac{y}{x^{2}+y^{2}}\) Path: y = 0 Points: (1, 0), (0.5, 0), (0.1, 0), (0.01, 0), (0.001, 0) Path: y = x Points: (1, 1), (0.5, 0.5), (0.1, 0.1), (0.01, 0.01), (0.001, 0.001) Text Transcription: (x, y) rightarrow (0,0) f(x, y)=y / x^2 + y^2

In Exercises 39–42, use a graphing utility to make a table showing the values of f(x, y) at the given points for each path. Use the result to make a conjecture about the limit of f(x, y) as \((x, y) \rightarrow(0,0)\). Determine whether the limit exists analytically and discuss the continuity of the function. \(f(x, y)=-\frac{x y^{2}}{x^{2}+y^{4}}\) Path: \(x=y^{2}\) Points: (1, 1), (0.25, 0.5), (0.01, 0.1), (0.0001, 0.01), (0.000001, 0.001) Path: \(x=-y^{2}\) Points: (-1, 1), (-0.25, 0.5), (-0.01, 0.1), (-0.0001, 0.01), (-0.000001, 0.001) Text Transcription: (x, y) rightarrow (0,0) f(x, y)=-xy^2 / x^2 + y^4 x=y^2 x=-y^2

In Exercises 39–42, use a graphing utility to make a table showing the values of f(x, y) at the given points for each path. Use the result to make a conjecture about the limit of f(x, y) as \((x, y) \rightarrow(0,0)\). Determine whether the limit exists analytically and discuss the continuity of the function. \(f(x, y)=\frac{2 x-y^{2}}{2 x^{2}+y}\) Path: y = 0 Points: (1, 0), (0.25, 0), (0.01, 0), (0.001, 0.), (0.000001, 0) Path: y = x Points: (1, 1), (0.25, 0.5), (0.01, 0.01), (0.001, 0.001), (0.0001, 0.0001) Text Transcription: (x, y) rightarrow (0,0) f(x, y)=2x-y^2 / 2x^2 + y

In Exercises 43– 46, discuss the continuity of the functions f and g. Explain any differences. \(f(x, y)=\left\{\begin{array}{ll}\frac{x^{4}-y^{4}}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\0, & (x, y)=(0,0)\end{array}\right.\) \(g(x, y)=\left\{\begin{array}{ll}\frac{x^{4}-y^{4}}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\1, & (x, y)=(0,0)\end{array}\right.\) Text Transcription: f(x, y)={x^4 - y^4 / x^2 + y^2, & (x, y) neq (0,0) over 0, & (x, y)=(0,0) g(x, y)={x^4 - y^4 / x^2 + y^2, & (x, y) neq (0,0) over 1, & (x, y)=(0,0)

In Exercises 43– 46, discuss the continuity of the functions f and g. Explain any differences. \(f(x, y)=\left\{\begin{array}{ll}\frac{4 x^{4}-y^{4}}{2 x^{2}+y^{2}}, & (x, y) \neq(0,0) \\-1, & (x, y)=(0,0)\end{array}\right.\) \(g(x, y)=\left\{\begin{array}{ll}\frac{4 x^{4}-y^{4}}{2 x^{2}+y^{2}}, & (x, y) \neq(0,0) \\0, & (x, y)=(0,0)\end{array}\right.\) Text Transcription: f(x, y)={4x^4 - y^4 / 2x^2 + y^2, & (x, y) neq (0,0) over -1, & (x, y)=(0,0) g(x, y)={4x^4 - y^4 / 2x^2 + y^2, & (x, y) neq (0,0) over 0, & (x, y)=(0,0)

In Exercises 43– 46, discuss the continuity of the functions f and g. Explain any differences. \(f(x, y)=\left\{\begin{array}{ll}\frac{4 x^{2} y^{2}}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\0, & (x, y)=(0,0)\end{array}\right.\) \(g(x, y)=\left\{\begin{array}{ll}\frac{4 x^{2} y^{2}}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\2, & (x, y)=(0,0)\end{array}\right.\) Text Transcription: f(x, y)={4x^2 y^2 / x^2 + y^2, & (x, y) neq (0,0) over 0, & (x, y)=(0,0) g(x, y)={4x^2 y^2 / x^2 + y^2, & (x, y) neq (0,0) over 2, & (x, y)=(0,0)

In Exercises 43– 46, discuss the continuity of the functions f and g. Explain any differences. \(f(x, y)=\left\{\begin{array}{ll}\frac{x^{2}+2 x y^{2}+y^{2}}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\0, & (x, y)=(0,0)\end{array}\right.\) \(g(x, y)=\left\{\begin{array}{ll}\frac{x^{2}+2 x y^{2}+y^{2}}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\1, & (x, y)=(0,0)\end{array}\right.\) Text Transcription: f(x, y)={x^2 + 2xy^2 + y^2 / x^2 + y^2, & (x, y) neq (0,0) over 0, & (x, y)=(0,0) g(x, y)={x^2 + 2xy^2 + y^2 / x^2 + y^2, & (x, y) neq (0,0) over 1, & (x, y)=(0,0)

In Exercises 47–52, use a computer algebra system to graph the function and find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) (if it exists). f(x, y)=sin x + sin y Text Transcription: lim_(x, y) rightarrow (0,0) f(x, y)

In Exercises 47–52, use a computer algebra system to graph the function and find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) (if it exists). \(f(x, y)=\sin \frac{1}{x}+\cos \frac{1}{x}\) Text Transcription: lim_(x, y) rightarrow (0,0) f(x, y) f(x, y)=sin 1/x + cos 1/x

In Exercises 47–52, use a computer algebra system to graph the function and find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) (if it exists). \(f(x, y)=\frac{x^{2} y}{x^{4}+2 y^{2}}\) Text Transcription: lim_(x, y) rightarrow (0,0) f(x, y) f(x, y)=x^2 y / x^4 + 2y^2

In Exercises 47–52, use a computer algebra system to graph the function and find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) (if it exists). \(f(x, y)=\frac{x^{2}+y^{2}}{x^{2} y}\) Text Transcription: lim_(x, y) rightarrow (0,0) f(x, y) f(x, y)=x^2 + y^2 / x^2 y

In Exercises 47–52, use a computer algebra system to graph the function and find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) (if it exists). \(f(x, y)=\frac{5 x y}{x^{2}+2 y^{2}}\) Text Transcription: lim_(x, y) rightarrow (0,0) f(x, y) f(x, y)=5xy / x^2 + 2y^2

In Exercises 47–52, use a computer algebra system to graph the function and find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) (if it exists). \(f(x, y)=\frac{6 x y}{x^{2}+y^{2}+1}\) Text Transcription: lim_(x, y) rightarrow (0,0) f(x, y) f(x, y)=6xy / x^2 + y^2 + 1

In Exercises 53–58, use polar coordinates to find the limit. [Hint: Let \(x=r \cos \theta\) and \(y=r \sin \theta\) and note that \((x, y) \rightarrow(0,0)\) implies \(r \rightarrow \mathbf{0}\).] \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y^{2}}{x^{2}+y^{2}}\) Text Transcription: x=r cos theta y=r sin theta (x, y) rightarrow (0,0) r rightarrow 0 lim_(x, y) rightarrow (0,0) xy^2 / x^2 + y^2

In Exercises 53–58, use polar coordinates to find the limit. [Hint: Let \(x=r \cos \theta\) and \(y=r \sin \theta\) and note that \((x, y) \rightarrow(0,0)\) implies \(r \rightarrow \mathbf{0}\).] \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}+y^{3}}{x^{2}+y^{2}}\) Text Transcription: x=r cos theta y=r sin theta (x, y) rightarrow (0,0) r rightarrow 0 lim_(x, y) rightarrow (0,0) x^3 + y^3 / x^2 + y^2

In Exercises 53–58, use polar coordinates to find the limit. [Hint: Let \(x=r \cos \theta\) and \(y=r \sin \theta\) and note that \((x, y) \rightarrow(0,0)\) implies \(r \rightarrow \mathbf{0}\).] \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y^{2}}{x^{2}+y^{2}}\) Text Transcription: x=r cos theta y=r sin theta (x, y) rightarrow (0,0) r rightarrow 0 lim_(x, y)\rightarrow (0,0) x^2 y^2 / x^2 + y^2

In Exercises 53–58, use polar coordinates to find the limit. [Hint: Let \(x=r \cos \theta\) and \(y=r \sin \theta\) and note that \((x, y) \rightarrow(0,0)\) implies \(r \rightarrow \mathbf{0}\).] \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{\sqrt{x^{2}+y^{2}}}\) Text Transcription: x=r cos theta y=r sin theta (x, y) rightarrow (0,0) r rightarrow 0 lim_(x, y) rightarrow (0,0) x^2 - y^2 / sqrt x^2 + y^2

In Exercises 53–58, use polar coordinates to find the limit. [Hint: Let \(x=r \cos \theta\) and \(y=r \sin \theta\) and note that \((x, y) \rightarrow(0,0)\) implies \(r \rightarrow \mathbf{0}\).] \(\lim _{(x, y) \rightarrow(0,0)} \cos \left(x^{2}+y^{2}\right)\) Text Transcription: x=r cos theta y=r sin theta (x, y) rightarrow (0,0) r rightarrow 0 lim_(x, y) rightarrow (0,0) cos (x^2 + y^2)

In Exercises 53–58, use polar coordinates to find the limit. [Hint: Let \(x=r \cos \theta\) and \(y=r \sin \theta\) and note that \((x, y) \rightarrow(0,0)\) implies \(r \rightarrow \mathbf{0}\).] \(\lim _{(x, y) \rightarrow(0,0)} \sin \sqrt{x^{2}+y^{2}}\) Text Transcription: x=r cos theta y=r sin theta (x, y) rightarrow (0,0) r rightarrow 0 lim_(x, y) rightarrow (0,0) sin sqrt x^2 + y^2

In Exercises 59– 62, use polar coordinates and L’Hôpital’s Rule to find the limit. \(\lim _{(x, y) \rightarrow(0,0)} \frac{\sin \sqrt{x^{2}+y^{2}}}{\sqrt{x^{2}+y^{2}}}\) Text Transcription: lim_(x, y) rightarrow (0,0) sin sqrt x^2 + y^2 / sqrt x^2 + y^2

In Exercises 59– 62, use polar coordinates and L’Hôpital’s Rule to find the limit. \(\lim _{(x, y) \rightarrow(0,0)} \frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}\) Text Transcription: lim_(x, y) rightarrow (0,0) sin (x^2 + y^2) / x^2 + y^2

In Exercises 59– 62, use polar coordinates and L’Hôpital’s Rule to find the limit. \(\lim _{(x, y) \rightarrow(0,0)} \frac{1-\cos \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}\) Text Transcription: lim_(x, y) rightarrow (0,0) 1-cos (x^2 + y^2) / x^2 + y^2

In Exercises 59– 62, use polar coordinates and L’Hôpital’s Rule to find the limit. \(\lim _{(x, y) \rightarrow(0,0)}\left(x^{2}+y^{2}\right) \ln \left(x^{2}+y^{2}\right)\) Text Transcription: lim_(x, y) rightarrow (0,0) (x^2 + y^2) ln (x^2 + y^2)

In Exercises 63– 68, discuss the continuity of the function. \(f(x, y, z)=\frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}\) Text Transcription: f(x, y, z)=1 / sqrt x^2 + y^2 + z^2

In Exercises 63– 68, discuss the continuity of the function. \(f(x, y, z)=\frac{z}{x^{2}+y^{2}-4}\) Text Transcription: f(x, y, z)=z / x^2 + y^2 - 4

In Exercises 63– 68, discuss the continuity of the function. \(f(x, y, z)=\frac{\sin z}{e^{x}+e^{y}}\) Text Transcription: f(x, y, z)=sin z / e^x + e^y

In Exercises 63– 68, discuss the continuity of the function. f(x, y, z)=xy sin z

In Exercises 63– 68, discuss the continuity of the function. \(f(x, y)=\left\{\begin{array}{l}\frac{\sin x y}{x y}, \quad x y \neq 0 \\1, \quad x y=0\end{array}\right.\) Text Transcription: f(x, y)={sin xy / xy, x y neq 0 over 1, xy=0

In Exercises 63– 68, discuss the continuity of the function. \(f(x, y)=\left\{\begin{array}{l}\frac{\sin \left(x^{2}-y^{2}\right)}{x^{2}-y^{2}}, \quad x^{2} \neq y^{2} \\1, \quad x^{2}=y^{2}\end{array}\right.\) Text Transcription: f(x, y)={sin (x^2 - y^2) / x^2 - y^2, x^2 neq y^2 over 1, x^2 = y^2

In Exercises 69–72, discuss the continuity of the composite function \(f \circ g\). \(f(t)=t^{2}\) g(x, y)=2x - 3y Text Transcription: f circ g f(t)=t^2

In Exercises 69–72, discuss the continuity of the composite function \(f \circ g\). \(f(t)=\frac{1}{t}\) \(g(x, y)=x^{2}+y^{2}\) Text Transcription: f circ g f(t)=1/t g(x, y)=x^2 + y^2

In Exercises 69–72, discuss the continuity of the composite function \(f \circ g\). \(f(t)=\frac{1}{t}\) g(x, y)=2x - 3y Text Transcription: f circ g f(t)=1/t

In Exercises 69–72, discuss the continuity of the composite function \(f \circ g\). \(f(t)=\frac{1}{1-t}\) \(g(x, y)=x^{2}+y^{2}\) Text Transcription: f circ g f(t)=1 / 1-t g(x, y)=x^2 + y^2

In Exercises 73–78, find each limit. (a) \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) (b) \(\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\) \(f(x, y)=x^{2}-4 y\) Text Transcription: lim_Delta x rightarrow 0 f(x+Delta x, y) - f(x, y) / Delta x lim_Delta y rightarrow 0 f(x, y+Delta y) - f(x, y) / Delta y f(x, y)=x^2 - 4y

In Exercises 73–78, find each limit. (a) \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) (b) \(\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\) \(f(x, y)=x^{2}+y^{2}\) Text Transcription: lim_Delta x rightarrow 0 f(x+Delta x, y) - f(x, y) / Delta x lim_Delta y rightarrow 0 f(x, y+Delta y) - f(x, y) / Delta y f(x, y)=x^2 + y^2

In Exercises 73–78, find each limit. (a) \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) (b) \(\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\) \(f(x, y)=\frac{x}{y}\) Text Transcription: lim_Delta x rightarrow 0 f(x+Delta x, y) - f(x, y) / Delta x lim_Delta y rightarrow 0 f(x, y+Delta y) - f(x, y) / Delta y f(x, y)=x/y

In Exercises 73–78, find each limit. (a) \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) (b) \(\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\) \(f(x, y)=\frac{1}{x+y}\) Text Transcription: lim_Delta x rightarrow 0 f(x+Delta x, y) - f(x, y) / Delta x lim_Delta y rightarrow 0 f(x, y+Delta y) - f(x, y) / Delta y f(x, y)=1 / x+y

In Exercises 73–78, find each limit. (a) \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) (b) \(\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\) f(x, y)=3x + xy - 2y Text Transcription: lim_Delta x rightarrow 0 f(x+Delta x, y) - f(x, y) / Delta x lim_Delta y rightarrow 0 f(x, y+Delta y) - f(x, y) / Delta y

In Exercises 73–78, find each limit. (a) \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}\) (b) \(\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\) \(f(x, y)=\sqrt{y}(y+1)\) Text Transcription: lim_Delta x rightarrow 0 f(x+Delta x, y) - f(x, y) / Delta x lim_Delta y rightarrow 0 f(x, y+Delta y) - f(x, y) / Delta y f(x, y)=sqrt y (y+1)

True or False? In Exercises 79–82, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=0\), then \(\lim _{x \rightarrow 0} f(x, 0)=0\). Text Transcription: lim_(x, y) rightarrow (0,0) f(x, y)=0 lim_x rightarrow 0 f(x, 0)=0

True or False? In Exercises 79–82, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\lim _{(x, y) \rightarrow(0,0)} f(0, y)=0\), then \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=0\). Text Transcription: lim_(x, y) rightarrow (0,0) f(0, y)=0 lim_(x, y) rightarrow (0,0) f(x, y)=0

True or False? In Exercises 79–82, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f is continuous for all nonzero x and y, and f(0,0)=0, then \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=0\). Text Transcription: lim_(x, y) rightarrow (0,0) f(x, y)=0

True or False? In Exercises 79–82, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If g and h are continuous functions of x and y, and f(x, y) = g(x)+h(y), then f is continuous.

Consider \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{x y}\) (see figure). (a) Determine (if possible) the limit along any line of the form y = ax. (b) Determine (if possible) the limit along the parabola \(y=x^{2}\). (c) Does the limit exist? Explain. Text Transcription: lim_(x, y) rightarrow (0,0) x^2 + y^2 / xy) y=x^2

Consider \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y}{x^{4}+y^{2}}\) (see figure). (a) Determine (if possible) the limit along any line of the form y = ax. (b) Determine (if possible) the limit along the parabola \(y=x^{2}\). (c) Does the limit exist? Explain. Text Transcription: lim_(x, y) rightarrow (0,0) x^2 y / x^4 + y^2 y=x^2

In Exercises 85 and 86, use spherical coordinates to find the limit. [Hint: Let \(x=\rho \sin \phi \cos \theta\), \(y=\rho \sin \phi \sin \theta\) and \(z=\rho \cos \phi\), and \((x, y, z) \rightarrow(\mathbf{0}, \mathbf{0}, \mathbf{0})\) note that implies \(\rho \rightarrow \mathbf{0}^{+}\). ] \(\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x y z}{x^{2}+y^{2}+z^{2}}\) Text Transcription: x=rho sin phi cos theta y=rho sin phi sin theta z=rho cos phi (x, y, z) rightarrow (0, 0, 0) rho rightarrow 0^+ lim_(x, y, z) rightarrow (0,0,0) xyz / x^2 + y^2 + z^2

In Exercises 85 and 86, use spherical coordinates to find the limit. [Hint: Let \(x=\rho \sin \phi \cos \theta\), \(y=\rho \sin \phi \sin \theta\) and \(z=\rho \cos \phi\), and \((x, y, z) \rightarrow(\mathbf{0}, \mathbf{0}, \mathbf{0})\) note that implies \(\rho \rightarrow \mathbf{0}^{+}\). ] \(\lim _{(x, y, z) \rightarrow(0,0,0)} \tan ^{-1}\left[\frac{1}{x^{2}+y^{2}+z^{2}}\right]\) Text Transcription: x=rho sin phi cos theta y=rho sin phi sin theta z=rho cos phi (x, y, z) rightarrow (0, 0, 0) rho rightarrow 0^+ lim_(x, y, z) rightarrow (0,0,0) tan^-1 [1 / x^2 + y^2 + z^2]

Find the following limit. \(\lim _{(x, y) \rightarrow(0,1)} \tan ^{-1}\left[\frac{x^{2}+1}{x^{2}+(y-1)^{2}}\right]\) Text Transcription: lim_(x, y) rightarrow (0,1) tan^-1 [x^2 + 1 / x^2 + (y-1)^2]

For the function \(f(x, y)=x y\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)\) define f(0, 0) such that f is continuous at the origin. Text Transcription: f(x, y)=xy (x^2 - y^2 / x^2 + y^2)

Prove that \(\lim _{(x, y) \rightarrow(a, b)}[f(x, y)+g(x, y)]=L_{1}+L_{2}\) where f(x, y) approaches \(L_{1}\) and g(x, y) approaches \(L_{2}\) as \((x, y) \rightarrow(a, b)\). Text Transcription: lim_(x, y) rightarrow (a, b) [f(x, y)+g(x, y)] = L_1 + L_2 L_1 L_2 (x, y) rightarrow (a, b)

Prove that if f is continuous and f(a, b) < 0, there exists a \(\delta\)-neighborhood about (a, b) such that f(a, b) < 0 for every point (x, y) in the neighborhood. Text Transcription: delta

Define the limit of a function of two variables. Describe a method for showing that \(\lim _{(x, y) \rightarrow \left(x_{0}, y_{0}\right)} f(x, y)\) does not exist. Text Transcription: lim_(x, y) rightarrow (x_0, y_0) f(x, y)

State the definition of continuity of a function of two variables.

Determine whether each of the following statements is true or false. Explain your reasoning. (a) If \(\lim _{(x, y) \rightarrow(2,3)} f(x, y)=4\), then \(\lim _{x \rightarrow 2} f(x, 3)=4\). (b) If \(\lim _{x \rightarrow 2} f(x, 3)=4\), then \(\lim _{(x, y) \rightarrow(2,3)} f(x, y)=4\). (c) If \(\lim _{x \rightarrow 2} f(x, 3)=\lim _{y \rightarrow 3} f(2, y)=4\), then \(\lim _{(x, y) \rightarrow(2,3)} f(x, y)=4\). (d) If \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=0\), then for any real number k, \(\lim _{(x, y) \rightarrow(0,0)} f(k x, y)=0\). Text Transcription: lim_(x, y) rightarrow (2,3) f(x, y) = 4 lim_x rightarrow 2 f(x, 3) = 4 lim_x rightarrow 2} f(x, 3) = 4 lim_(x, y) rightarrow (2,3) f(x, y) = 4 lim_x rightarrow 2 f(x, 3) = lim_y rightarrow 3 f(2, y) = 4 lim_(x, y) rightarrow (2,3) f(x, y) = 4 lim_(x, y) rightarrow (0,0) f(x, y) = 0 lim_(x, y) rightarrow (0,0) f(k x, y) = 0

(a) If f(2, 3) = 4, can you conclude anything about \(\lim _{(x, y) \rightarrow(2,3)} f(x, y)\)? Give reasons for your answer. (b) If \(\lim _{(x, y) \rightarrow(2,3)} f(x, y)=4\), can you conclude anything about f(2, 3)? Give reasons for your answer. Text Transcription: lim_(x, y) rightarrow (2,3) f(x, y) lim_(x, y) rightarrow (2,3) f(x, y) = 4