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(a) Let f : R- Rbe function defined by f () sin % Then f is continous at = [ Hinl. Use a trig: Identity below: For any 01, 02 € R; 01 + 02 01 02 sin 0 1 sin 0...

Question

(a) Let f : R- Rbe function defined by f () sin % Then f is continous at = [ Hinl. Use a trig: Identity below: For any 01, 02 € R; 01 + 02 01 02 sin 0 1 sin 02 = 2 cOS Sin

(a) Let f : R- Rbe function defined by f () sin % Then f is continous at = [ Hinl. Use a trig: Identity below: For any 01, 02 € R; 01 + 02 01 02 sin 0 1 sin 02 = 2 cOS Sin



Answers

Let $$f(x, y)=\left\{\begin{array}{ll} \frac{1+2 x y-\cos x y}{x y} & \text { if } x y \neq 0 \\ a & \text { if } x y=0 \end{array}\right.$$ Find the value of $a$ for which $f$ is continuous at all points in $\mathbb{R}^{2}$.

Here we are going purse. Our problem. About 10 years here we have to prove that half of excess continues at cereal commerce. It'll so function f of X going away because, saying x squared plus nice square, different by X squared, plus life where floor explain nor tinkler, too. Was it a little similarly half off. Excluding late equals, learn for take flight equals zero comma little. So only find the limit for a further point. There Accomazzo and if exists them to function. It's candy nose at zero comma. Zero. So living X Come on, Lee things to their uncle. My little that function off externally, Mrs Carter, limit picks going away. And still there are commas side and describe. Plus why square they were hurt by X squared plus y squared for luck. Is there a reason Carter? It was Claire plus nice square. Prove it. Excluding away things to their common little. Then they turn stoop. So what will be getting is like limit their bench to zero. Saying that donor base there, which is close to learn so far can it go if they don't limit excluding away stoop close. Yeah, excludable that so half off X rays continues. They're going a little

So the given value is here effect says signed by Yaks And below side is X and one -X. So it X equals to zero value will be F0 will be Exact value of zero. Yes This value becomes zero So it will be infinitely F1 also Swiss infinity now. So it is saying define F of zero and therefore one so that the function is continuous in interval 0 to 1. Open Bracket 0: one. So this would be required. This is not defined, it function mhm Not defined. This is at 0 -0 0 And this is also at 1 0 plus and one define act zero and one. Mhm. So F zero should be infinity and F of one exact so also the infinity to make conscience continuous at and point continuous in 0- one day. This is our solution.

Well here the funds and detainees Fx says signed by X important X 1- sets this we can also write us. I know by X it's minus X squared right So limit x approaches zero san kayaks. X minus expert can be detainers using l hospitals. We can differentiate Limit X approaches zero. It can be written as course of antibiotics into type And the below difference. Yes and is 1 -1 two works right? Putting the value we get also by in two pi Course of zero. Sorry, expertly zero 1 -2 in 20 which is Buy a phone one you can save by. Mhm So F of zero should be bye. And Limit x approaches one signed by attacks upon X minus X where we can also write us again using a lot of hospitals, Limit x approaches one horse by X By upon 1 -2 stripe. So limit x approaches one. Just put down the value 1 -2 And the upside will be -1 in to buy, which will become yeah minus why. Or we can say again, the value becomes hi, so half of one should also be by so these two are solution.

A function off very well. X Way is given as sine x x squared plus y squared minus one Do very good Bye x X squared plus y squared minus one. If x x squared plus y squared is not equal to one and x x squared Plus y is well is equal to one. So solving this equation and to find out the value of V by the matter off limit V C There function F x y is equal toe limit off. X y is sine x x squared plus y squared minus one divided way excess square Phyllis by X squared minus V In the standard form, we know their sign ex upon X. When limit excess standing zero comes out as went. So the whole function will value off one as the limit off the function X by is one so value off we will be. One function will be continues at this point. So this is the answer off the question


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