the-sandwich-theorem-for-functions-of-two-variables-states-that-if-gx-y-leq-fx-y-leq-hx-y-for-all-11

Question

The Sandwich Theorem for functions of two variables states that if $g(x, y) \leq f(x, y) \leq h(x, y)$ for all $(x, y) 
eq\left(x_{0}, y_{0}ight)$ in a disk centered at $\left(x_{0}, y_{0}ight)$ and if $g$ and $h$ have the same finite limit $L$ as $(x, y) ightarrow\left(x_{0}, y_{0}ight),$ then
$$
\lim _{(x, y) ightarrow\left(x_{0}, y_{0}ight)} f(x, y)=L
$$
Use this result to support your answers to the questions in Exercises $45-48 .$
Does knowing that
$$
2|x y|-\frac{x^{2} y^{2}}{6}<4-4 \cos \sqrt{|x y|}<2|x y|
$$
tell you anything about
$$
\lim _{(x, y) ightarrow(0,0)} \frac{4-4 \cos \sqrt{|x y|}}{|x y|} ?
$$
Give reasons for your answer.

Uma Kumari · Accepted Answer

it has given that too, into eggs, while more minus X square into by squared, divided by six. The smaller than four minus four into co sign on the route takes in tow by mood, which is smaller than two in two weeks. Into my mood, divide the whole function by X into my mode. We get tu minus X squared into by square, divided by six into eggs into my mood, which is smaller than four minus four into co sign under Route X into y mode divided by eggs into my mood, which is greater than since X square in tow by square is equals two x into my whole square, which is equals toe X in tow by mood square. Then we get tu minus eggs into by more divided by assets, which is smaller than four minus four in Tokyo sign under Route It seemed to buy more, divided by its into my mood, it just smaller than two now taking limit as X coma vitamins to zero comma zero. We get Lim X comma vitamins to zero comma zero tu minus X tinto by more divided by six, which is smaller than limit extent X comma vitamins to zero comma zero four minus four into casino under Route X into y more divided by X and y mode, which is smaller than Lim X coma. Vitamins to zero comma zero. So since limit ex coma vitamins to zero comma. Zero tu minus eggs into my mood, divided by states as equals toe minus zero, which is equals toe on lim X comma. Vitamins to zero comma. Zero. Who is equals to do so best Sandwich theorem We have limit X comma vitamins to zero comma zero four minus four into casino under root eggs into by more divided by eggs into my mode. There&#x27;s equals to do. This is the answer.

Brittany Knowlton · Answer

So for this problem, it gives us a CE long inequality, everyone a share. What? There&#x27;s knowing that this equality tell us anything about the limit. Foremost for coastlines groove X y over x y and then to give reasons for answer. So before we go into this problem, it does give us the sandwich there. Um and it tells us to use this result in support. Answer in this question. So I&#x27;m going to go ahead and just rewrite the sandwich there. Um, so getting sandwich Sam. And that tells us if g of x y is listener equal to f x y, which is less angled h of X y and G and H have same limit l as X y those two ex not Why not, Then the limit as X y goes to ex not why not of that middle term. So another gene in age both goto el as X y goes to explain why not in the limit from the same of f is also going to equal else again. Kind of appropriately named sandwich there. Let me go sell lim go sell, then inside a sandwich also go stow. So first question, we&#x27;re giving this inequality two X Why minus X squared y swears over six less than four minus for co sign of the square root of That&#x27;s about you, x y. It&#x27;s less than tio absolute value of X Y. And we want to know what this can tell us about the limit and X Y goes to 00 of four minus for co sign zero x y over. Excellent. So we&#x27;re gonna do this by using the sandwich there. So I want to have this term in the middle and find the limits of either side. So the only thing I&#x27;m missing is that this needs to be divided by the absolute value of X Y. I&#x27;m going to go ahead and do that here. And so when I do that, this just becomes to minus X squared y squared over six times a Valium x y less than four minus for co sign. Thanks. Why over times the value of X y Lesson two so clearly on the right hand side, this limit as X Y goes to 00 up to is obviously unequal, Tio. So I really want this limits equal to as well. So I take this limit. I have to be careful. There is an absolute value are here. So I&#x27;m gonna have to go about two ways. Looking at first when x Y is less than zero and then when x y is greater than zero. So 1st 1 d&#x27;oh! When X Y is greater than zero then I&#x27;m just going to get plummet. X y goes to 00 a cz two minus. Since this is positive, that means I&#x27;m just The sign stays the same And then I can cancel out on top of bottoms. I get X y over six and I&#x27;m taking it when expected. Zeros This tremor there is out and I get to And then when I am looking at it when x y it&#x27;s lesson zero, then just becomes negative. So this is when I switched o positive. And so then the x y over six, and that&#x27;s gonna change anything. It&#x27;s still going to be equal to two. So I have the limit on the left hand side goes to the limit and the right hand side close to two. So that means this limit by the sandwich dam is going to equal Tio

Ahmed Ibrahim · Answer

in this question, we will start boy taking X and y graters and zero so hour limit. Um off the function will be equal. We can substitute absolute X and y by post of X and y so we will have to x y minus excess square y square over six over posted X y Now we can simplify x y you is x y and x y so we can write it as two minus X y over six over one Bush we can&#x27;t move and this will be our solution. Then we can now substitute X and Y boy zero. So our final solution will be equal to then the other limit or other age. When X and y approaches zero, we can simplify it as as we say, toe multiply X on Roy postive over x y post of we can dream in X y is X y so are fine. Absolution will be toe then the other option when X and y is less than zero so hour limit. In this case, we can substitute X and y We used negative external or absolute X and y used negative son y. So we will have negative two X Y minus X squared y square over six over negative ex watch Also, now we can remove negative X y with negative X y with negative X y So our solution will be to plus X y over six in this case. Also, if we substitute X and Y by zero so our final solution will be to and for the second edge or the other age when X and Y approaches zero, we can substitute it as negative tool X Y and negative X y So we can simplify. Negative exploring is negative and X y so. Also our solution with people finally. So the limit off our function when X and y approaches zero will be equal to according or by using the thunder. Which team? Thank you.

Uma Kumari · Answer

it is given that co signed into one upon by mode smaller than equals to one we need to find limit. It&#x27;s common by tends to zero comma zero extent to co sign into one upon by the limit is zero sense Cosigned man upon by mold is smaller than equals to one. We know that a science data lies between minus one and one, so we begin minus one is smaller than equals To consign integon upon by which is smaller than equals toe one might apply the whole function by AEG speak it minus X, which is smaller than equals toe eggs into designing toe one upon by which is smaller than equals two way four x s greater than equals to zero For access Mulligan equals to zero minus X is greater than equals two weeks into co sign into one upon by which is greater than equal straight. So we take the limits off minus X and Eggs LTD&#x27;s comma via tends to zero comma zero minus x toe get zero and lim X coma via tends to zero comma zero. Okay, get zero by the sandwich there, um, we get g o h comma by a smaller than equals to a perfect storm of I, which is also smaller than equals to a topics comma. Then Lim X comma vitamins to zero comma zero is extent to co sign into one upon by also tends to zero. Does the required solution is zero. This is done so

Uma Kumari · Answer

it is given that signing toe gonna phone eggs small A smaller than equals toe one that is minus one is smaller than equals to sign into one upon X which is smaller than equals to one. My deployed the whole function by why we get minus wire. Smaller than equals Toe Ryan to signing into Van upon X, which is smaller than equals to buy. Now taking the limit as Exco movie tends to zero comma zero v gate lim X coma via tends to zero comma zero minus y is smaller than equals to limit next coma by tends to zero comma zero buying to sign into one opponent which is smaller than equals toe limit ex coma vitamins to zero comma zero Why since lament It&#x27;s common violence to zero comma zero minus y is equals to limit extent X comma violence to zero comma zero vice equals to zero. So by using the family care, um, we must have lim x comma vitamins to zero comma zero buying to sign into one upon X as equals to zero. This is done so

Brittany Knowlton · Answer

So first question it says, Does knowing that coastline of one over wide ups the value that lasts? Two people the one tell you anything about the limit of X Times Co sign of one over why and then to give reasons for your answer and also the beginning of the section. It did tell us to use the sandwich There result for these questions so that going into this week, first R looking at that, we know co sign of one over why it&#x27;s lost wrinkled one. We want to see what this tells us about Clement as X y 00 of extremes co sign of one over. Why? So you wanted to tell us to use a sandwich there? Um, so I&#x27;m first going to make this into a long qualities. I can use that there. I have absolute value bars hears. This is really the same thing as negative Linus lesson able to co sign of one over why a single one and so sandwich there and tells me that if both of these outside when I take the limit of both these outside terms and so it goes to the same limit, then want to take that same limit on the inside term. It&#x27;s going to go to that same limousine well, sandwiching itself in. So I want to know the limit of X times this middle term. So to get that, I&#x27;m gonna multiply by X on both sides, and I see that I won&#x27;t have to different options All happen either. Have this. This is gonna happen when X is greater than zero or I&#x27;m gonna have Negative X is greater than equal to x co Sign of one over why Screening X Mrs happens when exes lesson zero. Either way, we still have the same term on either side. So taking the limit of the negative X looking for the same limit for both my steak. For negative X, I get zero. Want to take it for positive X a Soviet zero So I can use my sandwich there. I&#x27;m here. And so since both limits on the outside or equals zero, that means the limit on the inside. It&#x27;s been equal zero by sandwich. Damn

Problem

The Sandwich Theorem for functions of two variabl…

Problem 46 Hard Difficulty

Answer

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