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2 years ago

A. b. m c. m d. m e. m

Mathematics

1 answer:

just olya [345]2 years ago

5 0

I’m confused as to what the question is? if knew the question i could potentially help.

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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

Anyone know the middle question ?

jeyben [28]

Ab is obtuse acute to AD
DC is perpendicular parallel to AB

4 0

2 years ago

Read 2 more answers

The county fair charges $1.25 per ticket for the rides. Jermaine bought 25 tickets for the rides and spent a total of $43.75 at

satela [25.4K]

The total cost is the dependant variable and the number of tickets along with flat out price is the independent variable.

The equation would be y = 1.25x + ?

You would insert 25 for x which equals 31.25 and insert 43.75 for y.

43.75 = 31.25 + ? Then subtract 31.25 from both sides of the equation and you would get....

12.5 = ?

That is your flat rate to get in to the fair

7 0

3 years ago

How do you solve for x? -12+7=-5(x+2)

dusya [7]

-5= -5x-10

Plus 10 from both sides

5=-5x
X = -1

7 0

3 years ago

Read 2 more answers

Answer this plzzz my lil brother needs help

GREYUIT [131]

Answer:

6/8 or 3/4