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Question

If $f\left(x_{0}, y_{0}\right)=3,$ what can you say about
$$
(x, y) \rightarrow\left(x_{0}, y_{0}\right)
$$
if $f$ is continuous at $\left(x_{0}, y_{0}\right) ?$ If $f$ is not continuous at $\left(x_{0}, y_{0}\right) ?$ Give reasons for your answer.

Uma Kumari · Accepted Answer

suppose a face of function it is given that a four x note comma by not it&#x27;s equals 23 The pose if this continents at the point x not common by not the objective is to determine about the limit expression. Lim X Coma via things, too. It&#x27;s not comma. Why not? A 4 ft come over since FS continues at the point. X not comma by not definition of limit at the point implies that F is defined at x, not comma by not limit. Ex coma by tends to x Not common. Why not 1/4 ex coma? By exist, Andi equals toe 1/4 ex, not common. Why not? Therefore limit X com abidance to X Not common by note. A four X comma by is equals to afford Exco X not comma by note is equals two F or x, not comma by note, which is equals to three on. If the function if is not continues at its not common by not the limit could have any value different from three. Consider the following example A four x camera is equals toe X plus why divided by eggs coma X coma by is equals to zero comma zero on three government X comma y is not equals to zero comma zero If we move along the line y Z equals two eggs, then for old X is not equals to zero. 1/4 Exco my ex is equals two x plus x divided by two which is equals Toe X divided by two which is equals to do They are four. 1/4 ex coma by is equals toe as ex coma by is tends to zero comma zero along bicycles through it eggs but a 40 comma zero is equals to three.

Brittany Knowlton · Answer

looking at this question, something that we know is that the function Excellent wine ice equal to three. And we want to see what we can say about the limit as ex wise approaching this point of f of x y and two different cases. First, about this continuous at this point, and secondly, if it is not continuous, so for suppose this continuous no, it&#x27;s not. Then clearly the limit as X Y goes Teo. Excellent wine. I&#x27;m if X Y is equal to three and we just know this from technicians and everything. If it&#x27;s continuous there, then clearly the limit will just equal to three. Since we&#x27;re playing in these points for X and Y. However, if if it&#x27;s not continuous, then we don&#x27;t really know much about the value of the function. At that point, when we were taking the limit, the limit I could have any value different from three not really just depends on the type of this continuity there, but if you have any value different from three or the limit may not even exist at that point,

Ahmed Ibrahim · Answer

from this question. If the function f is continuous at explode and why no, then our limit when X and y approaches x Nord and why don&#x27;t for function f x on why must equal to function off ex moat on why node, which is equal to three as government.

Victoria Dollar · Answer

Okay, so we&#x27;re given this function of three sine X over X when X does not equal zero and then a when X equals zero and it&#x27;s asking us to find when is what value of a makes this function continuous. So what we want to do is we want to look at the limit when X is approaching zero for three sine X over X crypt. Now, what I can do since three is a constant, as I can pull three to the front. And now I&#x27;m just finding the limit of sine X over X and based off of the serum 3.10. It tells us that this here is actually equal toe one. So what I can do is three times one is going to equal three. So a would need to equal three in order for this function to be continuous

Sriram Soundarrajan · Answer

here we are going purse. Our problem. About 10 years here we have to prove that half of excess continues at cereal commerce. It&#x27;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&#x27;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&#x27;t limit excluding away stoop close. Yeah, excludable that so half off X rays continues. They&#x27;re going a little

Sarah Wharton · Answer

way have F of X is equal to X of the 2/3, and f prime of zero doesn&#x27;t exist, but it&#x27;s still continuous and need to prove that both graphically and numerically. So let&#x27;s prove that f of zero exists. F of zero is equal to zero to the 2/3 power, which is just equal to zero so f of zero definitely exists, which means you might think that it&#x27;s because it&#x27;s continuous. F f f of X is continuous at X equals zero that there wouldn&#x27;t be any problem calculating the derivative. But in fact, if you look at the graph of F of X is equal to X to the 2/3 power, you&#x27;ll see that there is a custom. So the left and right hand derivative are not equal. And that means that F prime of X is cannot be found at X equals zero. That&#x27;s it.

Problem

The Sandwich Theorem for functions of two variabl…

Problem 44 Medium Difficulty

If $f\left(x_{0}, y_{0}\right)=3,$ what can you say about
$$
(x, y) \rightarrow\left(x_{0}, y_{0}\right)
$$
if $f$ is continuous at $\left(x_{0}, y_{0}\right) ?$ If $f$ is not continuous at $\left(x_{0}, y_{0}\right) ?$ Give reasons for your answer.

Answer

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