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2 years ago

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A young women walked into a cafe and asked for a drink of water the man behind the counter suddenly pulled out a gun and pointed

it at her a few seconds elapsed and then the women smiled and thanked the man and left

1 answer:

dexar [7]2 years ago

7 0

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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

Ahat [919]

(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 $y=0$ .

5 0

1 year ago

HELP EASY POINTS!

aleksklad [387]

Answer:

y= 3x

Step-by-step explanation:

4 0

3 years ago

Does anyone know this?

Bond [772]

Step-by-step explanation:

$measure \: of \: interior \: angle \\ = \frac{ (18 - 2) \times 180 \degree}{18} \\ = 16 \times 10 \degree \\ = 160 \degree \\ \\ measure \: of \: exterior \: angle \\ = \frac{360 \degree}{18} = 20 \degree \\ \\ \\$

8 0

3 years ago

It takes 8 workers 12 days to build a wall work out how many days it would take 3 workers to complete the same job

MissTica

Answer: 4.5 days or 4 1/2 days

Step-by-step explanation: We can use cross product property, to solve this. Workers to Days remember that

6 0

2 years ago

5 more than 3 times a variable, x, is 23

Monica [59]

3x + 5 = 23

5 more than(addition) 3 times(multiplication) a variable(x) is(equals) 23

(you add 5 to 3x)

If you need to solve for "x", you need to isolate/get the variable "x" by itself in the equation:

3x + 5 = 23 Subtract 5 on both sides

3x + 5 - 5 = 23 - 5

3x = 18 Divide 3 on both sides to get "x" by itself

$\frac{3x}{3} =\frac{18}{3}$

x = 6

5 0

3 years ago