다변수함수의 극한과 연속

For the following exercises, use algebraic techniques to evaluate the limit.

$\lim_{(x,y)\to (0,0)}\frac{{x}^{4}-4{y}^{4}}{{x}^{2}+2{y}^{2}}$

For the following exercises, use algebraic techniques to evaluate the limit.

$\lim_{(x,y)\to (0,0)}\frac{{x}^{3}-{y}^{3}}{x-y}$

For the following exercises, use algebraic techniques to evaluate the limit.

$\lim_{(x,y)\to (0,0)}\frac{{x}^{2}-xy}{\sqrt{x}-\sqrt{y}}$

For the following exercises, evaluate the limits of the functions of three variables.

$\lim_{(x,y,z)\to (1,2,3)}\frac{x{z}^{2}-{y}^{2}z}{xyz-1}$

For the following exercises, evaluate the limits of the functions of three variables.

$\lim_{(x,y,z)\to (0,0,0)}\frac{{x}^{2}-{y}^{2}-{z}^{2}}{{x}^{2}+{y}^{2}-{z}^{2}}$

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.

$\lim_{(x,y)\to (0,0)}\frac{xy+{y}^{3}}{{x}^{2}+{y}^{2}}$

a. Along the $x\text{-axis}$ $(y=0)$ b. Along the $y\text{-axis}$ $(x=0)$ c. Along the path $y=2x$

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 $\lim_{(x,y)\to (0,0)}\frac{xy+{y}^{3}}{{x}^{2}+{y}^{2}}$ using the results of previous problem.

$\lim_{(x,y)\to (0,0)}\frac{{x}^{2}y}{{x}^{4}+{y}^{2}}$

a. Along the x-axis $(y=0)$ b. Along the y-axis $(x=0)$ c. Along the path $y={x}^{2}$

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 $\lim_{(x,y)\to (0,0)}\frac{{x}^{2}y}{{x}^{4}+{y}^{2}}$ using the results of previous problem.

Discuss the continuity of the following functions. Find the largest region in the $xy\text{-plane}$ 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 $xy\text{-plane}$ 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 $xy\text{-plane}$ in which the following functions are continuous.

$f(x,y)={e}^{3xy}$

Discuss the continuity of the following functions. Find the largest region in the $xy\text{-plane}$ in which the following functions are continuous.

$f(x,y)=\frac{1}{xy}$

For the following two exercises, determine the region in which the function is continuous. Explain your answer.

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

For the following two exercises, determine the region in which the function is continuous. Explain your answer.

$f(x,y)=\frac{\sin ({x}^{2}+{y}^{2})}{{x}^{2}+{y}^{2}}$

For the following two exercises, determine the region in which the function is continuous. Explain your answer.

Determine whether $f(x,y)$ is continuous at $(0,0)$. $f(x,y)=\{\begin{array}{cc}\frac{{x}^{2}y}{{x}^{2}+{y}^{2}} & \text{if}\,(x,y)\ne (0,0) \\ 0 & \text{if}\,(x,y)=(0,0)\end{array}\}$

For the following two exercises, determine the region in which the function is continuous. Explain your answer.

Determine whether $g(x,y)=\frac{{x}^{2}-{y}^{2}}{{x}^{2}+{y}^{2}}$ is continuous at $(0,0).$

Create a plot using graphing software to determine where the limit does not exist. Find where in the coordinate plane $f(x,y)=\frac{1}{{x}^{2}-y}$ is continuous.

Determine the region of the $xy\text{-plane}$ in which the function $g(x,y)=\arctan \left(\frac{x{y}^{2}}{x+y}\right)$ is continuous. Use technology to support your conclusion.

Determine the region of the $xy\text{-plane}$ in which $f(x,y)=\ln ({x}^{2}+{y}^{2}-1)$ is continuous. Use technology to support your conclusion. (Hint: Choose the range of values for $x\,\text{and}\,y$ carefully!)