
# 14.2E: Exercises for Section 14.2


1) Use the limit laws for functions of two variables to evaluate each limit below, given that $$\displaystyle \lim_{(x,y)→(a,b)}f(x,y) = 5$$ and $$\displaystyle \lim_{(x,y)→(a,b)}g(x,y) = 2$$.

1. $$\displaystyle \lim_{(x,y)→(a,b)}\left[f(x,y) + g(x,y)\right]$$
2. $$\displaystyle \lim_{(x,y)→(a,b)}\left[f(x,y) g(x,y)\right]$$
3. $$\displaystyle \lim_{(x,y)→(a,b)}\left[ \dfrac{7f(x,y)}{g(x,y)}\right]$$
4. $$\displaystyle \lim_{(x,y)→(a,b)}\left[\dfrac{2f(x,y) - 4g(x,y)}{f(x,y) - g(x,y)}\right]$$
1. $$\displaystyle \lim_{(x,y)→(a,b)}\left[f(x,y) + g(x,y)\right] = \displaystyle \lim_{(x,y)→(a,b)}f(x,y) + \displaystyle \lim_{(x,y)→(a,b)}g(x,y)= 5 + 2 = 7$$
2. $$\displaystyle \lim_{(x,y)→(a,b)}\left[f(x,y) g(x,y)\right] =\left(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)\right) \left(\displaystyle \lim_{(x,y)→(a,b)}g(x,y)\right) = 5(2) = 10$$
3. $$\displaystyle \lim_{(x,y)→(a,b)}\left[ \dfrac{7f(x,y)}{g(x,y)}\right] = \frac{7\left(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)\right)}{\displaystyle \lim_{(x,y)→(a,b)}g(x,y)}=\frac{7(5)}{2} = 17.5$$
4. $$\displaystyle \lim_{(x,y)→(a,b)}\left[\dfrac{2f(x,y) - 4g(x,y)}{f(x,y) - g(x,y)}\right] = \frac{2\left(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)\right) - 4 \left(\displaystyle \lim_{(x,y)→(a,b)}g(x,y)\right)}{\displaystyle \lim_{(x,y)→(a,b)}f(x,y) - \displaystyle \lim_{(x,y)→(a,b)}g(x,y)}= \frac{2(5) - 4(2)}{5 - 2} = \frac{2}{3}$$

In exercises 2 - 4, find the limit of the function.

2) $$\displaystyle \lim_{(x,y)→(1,2)}x$$

3) $$\displaystyle \lim_{(x,y)→(1,2)}\frac{5x^2y}{x^2+y^2}$$

$$\displaystyle \lim_{(x,y)→(1,2)}\frac{5x^2y}{x^2+y^2} = 2$$

4) Show that the limit $$\displaystyle \lim_{(x,y)→(0,0)}\frac{5x^2y}{x^2+y^2}$$ exists and is the same along the paths: $$y$$-axis and $$x$$-axis, and along $$y=x$$.

In exercises 5 - 19, 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.

5) $$\displaystyle \lim_{(x,y)→(0,0)}\frac{4x^2+10y^2+4}{4x^2−10y^2+6}$$

$$\displaystyle \lim_{(x,y)→(0,0)}\frac{4x^2+10y^2+4}{4x^2−10y^2+6} = \frac{2}{3}$$

6) $$\displaystyle \lim_{(x,y)→(11,13)}\sqrt{\frac{1}{xy}}$$

7) $$\displaystyle \lim_{(x,y)→(0,1)}\frac{y^2\sin x}{x}$$

$$\displaystyle \lim_{(x,y)→(0,1)}\frac{y^2\sin x}{x} = 1$$

8) $$\displaystyle \lim_{(x,y)→(0,0)}\sin(\frac{x^8+y^7}{x−y+10})$$

9) $$\displaystyle \lim_{(x,y)→(π/4,1)}\frac{y\tan x}{y+1}$$

$$\displaystyle \lim_{(x,y)→(π/4,1)}\frac{y\tan x}{y+1}=\frac{1}{2}$$

10) $$\displaystyle \lim_{(x,y)→(0,π/4)}\frac{\sec x+2}{3x−\tan y}$$

11) $$\displaystyle \lim_{(x,y)→(2,5)}(\frac{1}{x}−\frac{5}{y})$$

$$\displaystyle \lim_{(x,y)→(2,5)}(\frac{1}{x}−\frac{5}{y}) = −\frac{1}{2}$$

12) $$\displaystyle \lim_{(x,y)→(4,4)}x\ln y$$

13) $$\displaystyle \lim_{(x,y)→(4,4)}e^{−x^2−y^2}$$

$$\displaystyle \lim_{(x,y)→(4,4)}e^{−x^2−y^2} = e^{−32}$$

14) $$\displaystyle \lim_{(x,y)→(0,0)}\sqrt{9−x^2−y^2}$$

15) $$\displaystyle \lim_{(x,y)→(1,2)}(x^2y^3−x^3y^2+3x+2y)$$

$$\displaystyle \lim_{(x,y)→(1,2)}(x^2y^3−x^3y^2+3x+2y) = 11$$

16) $$\displaystyle \lim_{(x,y)→(π,π)}x\sin(\frac{x+y}{4})$$

17) $$\displaystyle \lim_{(x,y)→(0,0)}\frac{xy+1}{x^2+y^2+1}$$

$$\displaystyle \lim_{(x,y)→(0,0)}\frac{xy+1}{x^2+y^2+1} = 1$$

18) $$\displaystyle \lim_{(x,y)→(0,0)}\frac{x^2+y^2}{\sqrt{x^2+y^2+1}−1}$$

19) $$\displaystyle \lim_{(x,y)→(0,0)}\ln(x^2+y^2)$$

The limit does not exist because when $$x$$ and $$y$$ both approach zero, the function approaches $$\ln 0$$, which is undefined (approaches negative infinity).

In exercises 20 - 21, complete the statement.

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

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

Every open disk centered at $$(x_0,y_0)$$ contains points inside $$R$$ and outside $$R$$.

In exercises 22 - 25, use algebraic techniques to evaluate the limit.

22) $$\displaystyle \lim_{(x,y)→(2,1)}\frac{x−y−1}{\sqrt{x−y}−1}$$

23) $$\displaystyle \lim_{(x,y)→(0,0)}\frac{x^4−4y^4}{x^2+2y^2}$$

$$\displaystyle \lim_{(x,y)→(0,0)}\frac{x^4−4y^4}{x^2+2y^2} = 0$$

24) $$\displaystyle \lim_{(x,y)→(0,0)}\frac{x^3−y^3}{x−y}$$

25) $$\displaystyle \lim_{(x,y)→(0,0)}\frac{x^2−xy}{\sqrt{x}−\sqrt{y}}$$

$$\displaystyle \lim_{(x,y)→(0,0)}\frac{x^2−xy}{\sqrt{x}−\sqrt{y}} = 0$$

In exercises 26 - 27, evaluate the limits of the functions of three variables.

26) $$\displaystyle \lim_{(x,y,z)→(1,2,3)}\frac{xz^2−y^2z}{xyz−1}$$

27) $$\displaystyle \lim_{(x,y,z)→(0,0,0)}\frac{x^2−y^2−z^2}{x^2+y^2−z^2}$$

The limit does not exist.

In exercises 28 - 31, 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.

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

a. Along the $$x$$-axis $$(y=0)$$

b. Along the $$y$$-axis $$(x=0)$$

c. Along the path $$y=2x$$

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

The limit does not exist. The function approaches two different values along different paths.

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

a. Along the $$x$$-axis $$(y=0)$$

b. Along the $$y$$-axis $$(x=0)$$

c. Along the path $$y=x^2$$

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

The limit does not exist because the function approaches two different values along the paths.

In exercises 32 - 35, discuss the continuity of each function. Find the largest region in the $$xy$$-plane in which each function is continuous.

32) $$f(x,y)=\sin(xy)$$

33) $$f(x,y)=\ln(x+y)$$

The function $$f$$ is continuous in the region $$y>−x.$$

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

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

The function $$f$$ is continuous at all points in the $$xy$$-plane except at points on the $$x$$- and $$y$$-axes.

In exercises 36 - 38, determine the region in which the function is continuous. Explain your answer.

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

37) $$f(x,y)=$$$$\begin{cases}\dfrac{x^2y}{x^2+y^2} & if(x,y)≠(0,0)\\0 & if(x,y)=(0,0)\end{cases}$$

Hint:
Show that the function approaches different values along two different paths.
The function is continuous at $$(0,0)$$ since the limit of the function at $$(0,0)$$ is $$0$$, the same value of $$f(0,0).$$

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

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

The function is discontinuous at $$(0,0).$$ The limit at $$(0,0)$$ fails to exist and $$g(0,0)$$ does not exist.

40) Create a plot using graphing software to determine where the limit does not exist. Determine the region of the coordinate plane in which $$f(x,y)=\dfrac{1}{x^2−y}$$ is continuous.

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

Since the function $$\arctan x$$ is continuous over $$(−∞,∞), g(x,y)=\arctan(\frac{xy^2}{x+y})$$ is continuous where $$z=\dfrac{xy^2}{x+y}$$ is continuous. The inner function $$z$$ is continuous on all points of the $$xy$$-plane except where $$y=−x.$$ Thus, $$g(x,y)=\arctan(\frac{xy^2}{x+y})$$ is continuous on all points of the coordinate plane except at points at which $$y=−x.$$

42) Determine the region of the $$xy$$-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 $$x$$ and $$y$$ carefully!)

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

All points $$P(x,y,z)$$ in space

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

45) Show that $$\displaystyle \lim_{(x,y)→(0,0)}\frac{1}{x^2+y^2}$$ does not exist at $$(0,0)$$ by plotting the graph of the function.

The graph increases without bound as $$x$$ and $$y$$ both approach zero.

46) [T] Evaluate $$\displaystyle \lim_{(x,y)→(0,0)}\frac{−xy^2}{x^2+y^4}$$ by plotting the function using a CAS. Determine analytically the limit along the path $$x=y^2.$$

47) [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?

a.

b. The level curves are circles centered at $$(0,0)$$ with radius $$9−c$$.
c. $$x^2+y^2=9−c$$
d. $$z=3$$
e. $$\{(x,y)∈R^2∣x^2+y^2≤9\}$$
f. $$\{z|0≤z≤3\}$$

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

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

$$\displaystyle \lim_{(x,y)→(0,0)}\frac{\sin\sqrt{x^2+y^2}}{\sqrt{x^2+y^2}} = 1$$

50) Use polar coordinates to find $$\displaystyle \lim_{(x,y)→(0,0)}\cos(x^2+y^2).$$

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

$$f(g(x,y))$$ is continuous at all points $$(x,y)$$ that are not on the line $$2x−5y=0.$$

52) Given $$f(x,y)=x^2−4y,$$ find $$\displaystyle \lim_{h→0}\frac{f(x+h,y)−f(x,y)}{h}.$$

53) Given $$f(x,y)=x^2−4y,$$ find $$\displaystyle \lim_{h→0}\frac{f(1+h,y)−f(1,y)}{h}$$.

$$\displaystyle \lim_{h→0}\frac{f(1+h,y)−f(1,y)}{h} = 2$$