1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
jek_recluse [69]
1 year ago
5

Can you define f(0, 0) = c for some c that extends f(x, y) to be continuous at (0, 0)? If so, for what value of c? If not, expla

in why.​

Mathematics
1 answer:
Ahat [919]1 year ago
5 0

(i) Yes. Simplify f(x,y).

\displaystyle \frac{x^2 - x^2y^2 + y^2}{x^2 + y^2} = 1 - \frac{x^2y^2}{x^2 + y^2}

Now compute the limit by converting to polar coordinates.

\displaystyle \lim_{(x,y)\to(0,0)} \frac{x^2y^2}{x^2+y^2} = \lim_{r\to0} \frac{r^4 \cos^2(\theta) \sin^2(\theta)}{r^2} = 0

This tells us

\displaystyle \lim_{(x,y)\to(0,0)} f(x,y) = 1

so we can define f(0,0)=1 to make the function continuous at the origin.

Alternatively, we have

\dfrac{x^2y^2}{x^2+y^2} \le \dfrac{x^4 + 2x^2y^2 + y^4}{x^2 + y^2} = \dfrac{(x^2+y^2)^2}{x^2+y^2} = x^2 + y^2

and

\dfrac{x^2y^2}{x^2+y^2} \ge 0 \ge -x^2 - y^2

Now,

\displaystyle \lim_{(x,y)\to(0,0)} -(x^2+y^2) = 0

\displaystyle \lim_{(x,y)\to(0,0)} (x^2+y^2) = 0

so by the squeeze theorem,

\displaystyle 0 \le \lim_{(x,y)\to(0,0)} \frac{x^2y^2}{x^2+y^2} \le 0 \implies \lim_{(x,y)\to(0,0)} \frac{x^2y^2}{x^2+y^2} = 0

and f(x,y) approaches 1 as we approach the origin.

(ii) No. Expand the fraction.

\displaystyle \frac{x^2 + y^3}{xy} = \frac xy + \frac{y^2}x

f(0,y) and f(x,0) are undefined, so there is no way to make f(x,y) continuous at (0, 0).

(iii) No. Similarly,

\dfrac{x^2 + y}y = \dfrac{x^2}y + 1

is undefined when y=0.

You might be interested in
Solve for x.<br> 10 = x + 5 + x<br><br> A. 5<br> B. 10<br> C. 5/2 <br> D. no solution
Strike441 [17]
The answer is c
I love math btw
7 0
3 years ago
68°<br> radius=4.2cm<br> A sector of a circle is shown. Find its area, rounded to the nearest tenth.
andrezito [222]

Answer:

10

Step-by-step explanation:

8 0
3 years ago
Which theorem would you use to prove ABE ~ DCE?
Pachacha [2.7K]
Answer: Choice C) AA similarity

------------------------------------------------------------------------------
------------------------------------------------------------------------------

We are given that angle ABE = angle ECD since they are both right angles (square marker)

Angle AEB = angle CED as they are vertical angles

we have two pairs of congruent angles, leading us to be able to use the AA (angle angle) similarity theorem
3 0
3 years ago
Find the exponential function passing through points (-1,4/3) and (3,108)
guajiro [1.7K]

Answer:

its 152 i know cause i just did this....

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
A bag of 12 ounces of popcorn costs $4.80. What is the unit rate? (in others words what is the cost for one ounce) *​
masha68 [24]

Answer:

0.4

Step-by-step explanation:

$4.80 divided by 12 = 0.4

5 0
3 years ago
Read 2 more answers
Other questions:
  • Ḧ̤ë̤l̤̈p̤̈ p̤̈l̤̈ë̤ä̤s̤̈ë̤
    5·1 answer
  • Please helppp I suckk at fractions!!!
    13·1 answer
  • An artist is hired to create an art display for the interior of a city building. The display is to span a total width of 10 yd .
    5·1 answer
  • What is the solution of -3=r-4/6
    9·1 answer
  • If the radius of a circle is 33 what is the diameter?
    9·1 answer
  • A study is conducted to determine how long a person will wait on hold on a telephone call before hanging up. A sample of 1000 pe
    6·1 answer
  • This is a 2 answer question
    11·1 answer
  • I need help ASAP please help me
    8·1 answer
  • Lee kept track of how many kilometres she walked each week for the past two months.
    8·1 answer
  • Solve 3(b-5) = 27
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!