?For the following exercises, evaluate the limits at the indicated values of x and y. If

For the following exercises, find the limit of the function. \(\lim_{(x,\ y)\rightarrow(1,\ 2)}x\) Text Transcription: \lim_{(x,\ y)\rightarrow(1,\ 2)}x

For the following exercises, find the limit of the function. \(\lim_{(x,\ y)\rightarrow(1,\ 2)}\frac{5x^2y}{x^2+y^2}\) Text Transcription: \lim_{(x,\ y)\rightarrow(1,\ 2)}\frac{5x^2y}{x^2+y^2}

For the following exercises, find the limit of the function. Show that the limit \(\lim_{(x,\ y)\rightarrow(0,\ 0)}\frac{5x^2y}{x^2+y^2}\) exists and is the same along the paths: \(y\)-axis and \(x\)-axis, and along \(y\) = \(x\). Text Transcription: \lim_{(x,\ y)\rightarrow(0,\ 0)}\frac{5x^2y}{x^2+y^2} y-axis x-axis y = x

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(0,\ 0)}\frac{4x^2+10y^2+4}{4x^2-10y^2+6}\) Text Transcription: \lim_{(x,\ y)\rightarrow(0,\ 0)}\frac{4x^2+10y^2+4}{4x^2-10y^2+6}

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(11,\ 13)}\sqrt{\frac{1}{xy}}\) Text Transcription: \lim_{(x,\ y)\rightarrow(11,\ 13)}\sqrt{\frac{1}{xy}}\

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(0,\ 1)}\frac{y^2\sin x}{x}\) Text Transcription: \lim_{(x,\ y)\rightarrow(0,\ 1)}\frac{y^2\sin x}{x}

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(0,\ 0)}\sin\left(\frac{x^8+y^7}{x-y+10}\right)\) Text Transcription: \lim_{(x,\ y)\rightarrow(0,\ 0)}\sin\left(\frac{x^8+y^7}{x-y+10}\right)

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(\pi/4,\ 1)}\frac{y\tan x}{y+1}\) Text Transcription: \lim_{(x,\ y)\rightarrow(\pi/4,\ 1)}\frac{y\tan x}{y+1}

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \((x,\ y)\rightarrow(0,\ \pi/4)\frac{\sec x+2}{3x-\tan y}\) Text Transcription: (x,\ y)\rightarrow(0,\ \pi/4)\frac{\sec x+2}{3x-\tan y}

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(2,\ 5)}\left(\frac{1}{x}-\frac{5}{y}\right)\) Text Transcription: \lim_{(x,\ y)\rightarrow(2,\ 5)}\left(\frac{1}{x}-\frac{5}{y}\right)

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(4,\ 4)}x\ln y\) Text Transcription: \lim_{(x,\ y)\rightarrow(4,\ 4)}x\ln y

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(4,\ 4)}e^{-x^2-y^2}\) Text Transcription: \lim_{(x,\ y)\rightarrow(4,\ 4)}e^{-x^2-y^2}

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(0,\ 0)}\sqrt{9-x^2-y^2}\) Text Transcription: \lim_{(x,\ y)\rightarrow(0,\ 0)}\sqrt{9-x^2-y^2}

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim _{(x, y) \rightarrow(1,2)}\left(x^{2} y^{3}-x^{3} y^{2}+3 x+2 y\right)\) Text Transcription: \lim _{(x, y) \rightarrow(1,2)}\left(x^{2} y^{3}-x^{3} y^{2}+3 x+2 y\right)

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(\pi,\ \pi)}x\sin\left(\frac{x+y}{4}\right)\) Text Transcription: \lim_{(x,\ y)\rightarrow(\pi,\ \pi)}x\sin\left(\frac{x+y}{4}\right)

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(0,\ 0)}\frac{xy+1}{x^2+y^2+1}\) Text Transcription: \lim_{(x,\ y)\rightarrow(0,\ 0)}\frac{xy+1}{x^2+y^2+1}

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(0,\ 0)}\frac{x^2+y^2}{\sqrt{x^2+y^2+1}-1}\) Text Transcription: \lim_{(x,\ y)\rightarrow(0,\ 0)}\frac{x^2+y^2}{\sqrt{x^2+y^2+1}-1}

For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. \(\lim_{(x,\ y)\rightarrow(0,\ 0)}\ln\left(x^2+y^2\right)\) Text Transcription: \lim_{(x,\ y)\rightarrow(0,\ 0)}\ln\left(x^2+y^2\right)

For the following exercises, complete the statement. A point \(\left(x_{0}, y_{0}\right)\) in a plane region R is an interior point of R if ______________. Text Transcription: \left(x_{0}, y_{0}\right) R

For the following exercises, complete the statement. A point \(\left(x_{0}, y_{0}\right)\) in a plane region R is called a boundary point of R if _________. Text Transcription: \left(x_{0}, y_{0}\right)

For the following exercises, use algebraic techniques to evaluate the limit. \(\lim _{(x, y) \rightarrow(2,1)} \frac{x-y-1}{\sqrt{x-y}-1}\) Text Transcription: lim _(x, y) \rightarrow(2,1).frac.x-y-1/sqrt.x-y-1

For the following exercises, use algebraic techniques to evaluate the limit. \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) frac.x^4-4 y^4/x^2+2 y^2

For the following exercises, use algebraic techniques to evaluate the limit. \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}-y^{3}}{x-y} Text Transcription: lim _(x, y).rightarrow(0,0).frac.x^3-y^3/x-y

For the following exercises, use algebraic techniques to evaluate the limit. \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-x y}{\sqrt{x}-\sqrt{y}} Text Transcription: lim _(x, y) \rightarrow(0,0).frac.x^2-x y/sqrt.x-sqrt.y

For the following exercises, evaluate the limits of the functions of three variables. \(\lim _{(x, y, z) \rightarrow(1,2,3)} \frac{x z^{2}-y^{2} z}{x y z-1}\) Text Transcription: lim _(x, y, z)_rightarrow(1,2,3)_frac.x z^2-y^2 z/x y z-1

For the following exercises, evaluate the limits of the functions of three variables. \(\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x^{2}-y^{2}-z^{2}}{x^{2}+y^{2}-z^{2}}\) Text Transcription: lim _(x, y, z).rightarrow(0,0,0).frac.x^2-y^2-z^2/x^2+y^2-z^2

For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y+y^{3}}{x^{2}+y^{2}}\) a. Along the x-axis (y = 0) b. Along the y-axis (x = 0) c. Along the path y = 2x Text Transcription: lim _(x, y).rightarrow(0,0).frac.x y+y^3/x^2+y^2

For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. Evaluate \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y+y^{3}}{x^{2}+y^{2}}\) using the results of previous problem. Text Transcription: lim _(x, y).rightarrow(0,0).frac.x y+y^3/x^2+y^2

For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y}{x^{4}+y^{2}}\) a. Along the x-axis (y = 0) b. Along the y-axis (x = 0) c. Along the path \(y=x^{2}\) Text Transcription: lim _(x, y).rightarrow(0,0).frac.x^2 y/x^4+y^2 y=x^2

For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. Evaluate \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y}{x^{4}+y^{2}}\) using the results of previous problem. Text Transcription: lim _(x, y).rightarrow(0,0).frac.x^2 y/x^4+y^2

Discuss the continuity of the following functions. Find the largest region in the xy-plane in which the following functions are continuous. f (x, y) = sin(xy)

Discuss the continuity of the following functions. Find the largest region in the xy-plane in which the following functions are continuous. f (x, y) = ln(x + y)

Discuss the continuity of the following functions. Find the largest region in the xy-plane in which the following functions are continuous. \(f(x, y)=e^{3 x y}\) Text Transcription: f(x, y)=e^3 x y

Discuss the continuity of the following functions. Find the largest region in the xy-plane in which the following functions are continuous. \(f(x, y)=\frac{1}{x y}\) Text Transcription: f(x, y)=frac1/x y

For the following exercises, determine the region in which the function is continuous. Explain your answer. \(f(x, y)=\frac{x^{2} y}{x^{2}+y^{2}}\) Text Transcription: f(x, y)=frac.x^2y/x^2+y^2

For the following exercises, determine the region in which the function is continuous. Explain your answer. \(f(x, y)=\left\{\begin{array}{ll} \frac{x^{2} y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0) \end{array}\right\}\) (Hint: Show that the function approaches different values along two different paths.) Text Transcription: f(x, y)=left.begin.array.ll fracx^2y/x^2+y^2 & \text_if (x, y) neq(0,0) 0 & text { if }(x, y)=(0,0) end.array.right

For the following exercises, determine the region in which the function is continuous. Explain your answer. \(f(x, y)=\frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}\) Text Transcription: f(x, y)=frac.sin left(x^2+y^2right)/x^2+y^2

For the following exercises, determine the region in which the function is continuous. Explain your answer. Determine whether \(g(x, y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\) continuous at (0, 0). Text Transcription: g(x, y)=frac.x^2-y^2/x^2+y^2

For the following exercises, determine the region in which the function is continuous. Explain your answer. Create a plot using graphing software to determine where the limit does not exist. Determine the region of the coordinate plane in which \(f(x, y)=\frac{1}{x^{2}-y}\) is continuous. Text Transcription: f(x, y)=frac.1/x^2-y

For the following exercises, determine the region in which the function is continuous. Explain your answer. Determine the region of the xy-plane in which the composite function \(g(x, y)=\arctan \left(\frac{x y^{2}}{x+y}\right)\) is continuous. Use technology to support your conclusion. Text Transcription: g(x,y)=arctan_left(frac.x y^2/x+yright)

For the following exercises, determine the region in which the function is continuous. Explain your answer. Determine the region of the xy-plane in which \(f(x, y)=\ln \left(x^{2}+y^{2}-1\right)\) is continuous. Use technology to support your conclusion. (Hint: Choose the range of values for x and y carefully!) Text Transcription: f(x,y)=ln_left(x^2+y^2-1right)

For the following exercises, determine the region in which the function is continuous. Explain your answer. At what points in space is \(g(x, y, z)=x^{2}+y^{2}-2 z^{2}\) continuous? Text Transcription: g(x,y,z)=x^2+y^2-2 z^2

For the following exercises, determine the region in which the function is continuous. Explain your answer. At what points in space is \(g(x, y, z) = \frac{1}{x^{2}+z^{2}-1}\) continuous? Text Transcription: g(x,y,z)=frac.1/x^2+z^2-1

For the following exercises, determine the region in which the function is continuous. Explain your answer. Show that \(\lim _{(x, y) \rightarrow(0,0)} \frac{1}{x^{2}+y^{2}}\) does not exist at (0, 0) by plotting the graph of the function. Text Transcription: \lim _{(x,y)rightarrow(0,0)_frac1/x^2+y^2

For the following exercises, determine the region in which the function is continuous. Explain your answer. [T] Evaluate \(\lim _{(x, y) \rightarrow(0,0)} \frac{-x y^{2}}{x^{2}+y^{4}}\) by plotting the function using a CAS. Determine analytically the limit along the path \(x=y^{2}\). Text Transcription: lim _(x,y)rightarrow(0,0)_frac.-x y^2/x^2+y^4 x=y^2

For the following exercises, determine the region in which the function is continuous. Explain your answer. [T] a. Use a CAS to draw a contour map of \(z=\sqrt{9-x^{2}-y^{2}}\) b. What is the name of the geometric shape of the level curves? c. Give the general equation of the level curves. d. What is the maximum value of z? e. What is the domain of the function? f. What is the range of the function? Text Transcription: z=sqrt9-x^2-y^2

True or False: If we evaluate \(\lim _{(x, y) \rightarrow(0,0)} f(x)\) along several paths and each time the limit is 1, we can conclude that \(\lim _{(x, y) \rightarrow(0,0)} f(x)=1\) Text Transcription: lim _(x,y)rightarrow(0,0)_f(x) lim _(x,y)rightarrow(0,0)_f(x)=1

Use polar coordinates to find \(\lim _{(x, y) \rightarrow(0,0)} \frac{\sin \sqrt{x^{2}+y^{2}}}{\sqrt{x^{2}+y^{2}}}\) . You can also find the limit using L’Hôpital’s rule. Text Transcription: lim_(x, y) rightarrow (0,0) sin sqrt x^2 + y^2 / sqrt x^2 + y^2

Use polar coordinates to find \(\lim _{(x, y) \rightarrow(0,0)} \cos \left(x^{2}+y^{2}\right)\) . Text Transcription: lim_(x, y) rightarrow (0,0) cos (x^2 + y^2)

Discuss the continuity of f(g(x, y)) where f(t) = 1/t and g(x, y) = 2x ? 5y.

Given \(f(x, y)=x^{2}-4 y\), find \(\lim _{h \rightarrow 0} \frac{f(x+h, y)-f(x, y)}{h}\) . Text Transcription: f(x, y) = x^2 - 4y lim_h rightarrow 0 f(x+h, y) - f(x, y) / h

Given \(f(x, y)=x^{2}-4 y\), find \(\lim _{h \rightarrow 0} \frac{f(1+h, y)-f(1, y)}{h}\) . Text Transcription: f(x, y) = x^2 - 4y lim_h rightarrow 0 f(1+h, y) - f(1, y) / h