4.2 Limits and continuity  (Page 6/14)

$\underset{(x,y)\to (1,2)}{\text{lim}}x$

$\underset{(x,y)\to (1,2)}{\text{lim}}\frac{5x^{2}y}{x^2+y^2}$

Show that the limit $\underset{(x,y)\to (0,0)}{\text{lim}}\frac{5x^{2}y}{x^2+y^2}$ exists and is the same along the paths: $y\text{-axis}$ and $x\text{-axis,}$ and along $y=x.$

$\underset{(x,y)\to (0,0)}{\text{lim}}\frac{4x^2+10y^2+4}{4x^2-10y^2+6}$

$\underset{(x,y)\to (11,13)}{\text{lim}}\sqrt{\frac{1}{xy}}$

$\underset{(x,y)\to (0,1)}{\text{lim}}\frac{y^2\text{sin}x}{x}$

$\underset{(x,y)\to (0,0)}{\text{lim}}\text{sin}\left(\frac{x^8+y^7}{x-y+10}\right)$

$\underset{(x,y)\to (\pi \text{/}4,1)}{\text{lim}}\frac{y\text{tan}x}{y+1}$

$\underset{(x,y)\to (0,\pi \text{/}4)}{\text{lim}}\frac{\text{sec}x+2}{3x-\text{tan}y}$

$\underset{(x,y)\to (2,5)}{\text{lim}}\left(\frac{1}{x}-\frac{5}{y}\right)$

$\underset{(x,y)\to (4,4)}{\text{lim}}x\text{ln}y$

$\underset{(x,y)\to (4,4)}{\text{lim}}{e}^{\text{−}{x}^2-y^2}$

$\underset{(x,y)\to (0,0)}{\text{lim}}\sqrt{9-x^2-y^2}$

$\underset{(x,y)\to (1,2)}{\text{lim}}\left(x^2{y}^3-x^3{y}^2+3x+2y\right)$

$\underset{(x,y)\to (\pi ,\pi )}{\text{lim}}x\text{sin}\left(\frac{x+y}{4}\right)$

$\underset{(x,y)\to (0,0)}{\text{lim}}\frac{xy+1}{x^2+y^2+1}$

$\underset{(x,y)\to (0,0)}{\text{lim}}\frac{x^2+y^2}{\sqrt{x^2+y^2+1}-1}$

$\underset{(x,y)\to (0,0)}{\text{lim}}\text{ln}\left(x^2+y^2\right)$

A point $\left(x_0,y_0\right)$ in a plane region $R$ is an interior point of $R$ if _________________.

A point $\left(x_0,y_0\right)$ in a plane region $R$ is called a boundary point of $R$ if ___________.

$\underset{(x,y)\to (2,1)}{\text{lim}}\frac{x-y-1}{\sqrt{x-y}-1}$

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

$\underset{(x,y)\to (0,0)}{\text{lim}}\frac{x^3-y^3}{x-y}$

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

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

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

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

1. Along the $x\text{-axis}$ $(y=0)$
2. Along the $y\text{-axis}$ $(x=0)$
3. Along the path $y=2x$

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

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

1. Along the x -axis $(y=0)$
2. Along the y -axis $(x=0)$
3. Along the path $y=x^2$

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

$f(x,y)=\text{sin}(xy)$

$f(x,y)=\text{ln}(x+y)$

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

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

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

$f(x,y)=\left\{\begin{array}{cc}\frac{x^{2}y}{x^2+y^2}\hfill & \text{if}(x,y)\ne (0,0)\hfill \\ 0\hfill & \text{if}(x,y)=(0,0)\hfill \end{array}\right\}$

( Hint : Show that the function approaches different values along two different paths.)

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

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

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.

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

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

At what points in space is $g(x,y,z)=x^2+y^2-2z^2$ continuous?

At what points in space is $g(x,y,z)=\frac{1}{x^2+z^2-1}$ continuous?

Show that $\underset{(x,y)\to (0,0)}{\text{lim}}\frac{1}{x^2+y^2}$ does not exist at $\left(0,0\right)$ by plotting the graph of the function.

[T] Evaluate $\underset{(x,y)\to (0,0)}{\text{lim}}\frac{\text{−}x{y}^2}{x^2+y^4}$ by plotting the function using a CAS. Determine analytically the limit along the path $x=y^2.$

[T]

1. Use a CAS to draw a contour map of $z=\sqrt{9-x^2-y^2}.$
2. What is the name of the geometric shape of the level curves?
3. Give the general equation of the level curves.
4. What is the maximum value of $z?$
5. What is the domain of the function?
6. What is the range of the function?

True or False : If we evaluate $\underset{(x,y)\to (0,0)}{\text{lim}}f(x)$ along several paths and each time the limit is $1,$ we can conclude that $\underset{(x,y)\to (0,0)}{\text{lim}}f(x)=1.$

Use polar coordinates to find $\underset{(x,y)\to (0,0)}{\text{lim}}\frac{\text{sin}\sqrt{x^2+y^2}}{\sqrt{x^2+y^2}}.$ You can also find the limit using L’Hôpital’s rule.

Use polar coordinates to find $\underset{(x,y)\to (0,0)}{\text{lim}}\text{cos}\left(x^2+y^2\right).$

Discuss the continuity of $f(g(x,y))$ where $f(t)=1\text{/}t$ and $g(x,y)=2x-5y.$

Given $f(x,y)=x^2-4y,$ find $\underset{h\to 0}{\text{lim}}\frac{f(x+h,y)-f(x,y)}{h}.$

Given $f(x,y)=x^2-4y,$ find $\underset{h\to 0}{\text{lim}}\frac{f(1+h,y)-f(1,y)}{h}.$