the-sandwich-theorem-for-functions-of-two-variables-states-that-if-gx-y-leq-fx-y-leq-hx-y-for-all-13

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 $|\cos (1 / y)| \leq 1$ tell you anything about
$$
\lim _{(x, y) ightarrow(0,0)} x \cos \frac{1}{y} ?
$$
Give reasons for your answer.

Uma Kumari · Accepted 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

Uma Kumari · 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 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

Ahmed Ibrahim · Answer

in this question. On the limit is zero science. Absolute design. One where? Why Eastleigh Center Equal one. So we can write it as negative one is, then we&#x27;re equal. Co sign one over why they send or equal one. If we multiply by eggs, we can write it as negative X then or equal X the fine one over. Why the Centauri? Quint X, um for, uh, forex creators then or X greater than or equal to zero. But in case or four x the center or equals zero, we can write it as negative X creators then or equals X consigned one worldwide or eight hours, then or equal. Next is us as

Dillon Huddleston · Answer

s so we can indeed use the cellar stairs. In this case just to rewrite that you&#x27;ve in fact, about the sideboard of racks. Uh, is to say that this one Yeah, that&#x27;s people too. One of Rex isn&#x27;t saying what they want. Ah, now what we can say that our limit, which is going to be the absolute value of Ireland Georges didn&#x27;t like again. This is a of our function. Uh, why time sign with Rex. Eyes X, Y go to Georgia. This will be less than the product to limits. We&#x27;ll have the limit of why is why goes of the absolute value of y. That&#x27;s why in 20 uh, then multiply by one, we&#x27;ll be corresponding to the absolute value of someone Brax. And then if we why converse to zero? Uh, this implies the value of Ireland in the actual values are zero I wish is the same. We can think about it a cz we rewrote the bounding result on side of Sweetwater brags of ah saying that element iss great an equal to zero less wrinkled zero. This means that the very building then is indeed zero

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(Continuation of Example 4.) a. Reread Example 4…

Problem 48 Medium Difficulty

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