Squeeze Theorem - Multivarible question

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In summary, the student is having difficulty understanding how to find a limit when approaching (0,0). They are confused about how to set up the squeeze inequality. They are also unsure if they would need to use the squeeze theorem if they were doing the problem in polar coordinates.
  • #1
pbxed
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Hi,

I'm having a lot of difficulty with finding limits of multivariable functions. A question like this comes up every year in the final exam and it will always ask for use of the squeezing theorem.

Homework Statement



(a) Suppose that
f(x, y) = 1 +(5x2y3)/x2 + y2

for (x, y) =/= (0, 0)

and that f(0, 0) = 0. By applying the Squeezing Rule to |f(x, y) − 1|, or otherwise, prove

that f(x, y) -> 1 as (x, y) -> (0, 0).

Homework Equations





The Attempt at a Solution



I understand that in order for a limit to exist that no matter what direction we approach (0,0) we must compute the same value. From x-axis and y-axis it seems that the limit is indeed 1. I also get the intuition of squeeze theorem that

lim (x,y) -> (a,b) g(x) <= lim (x,y) -> (a,b) f(x) <= lim (x,y) -> (a,b) h(x)

so lim g(x) = lim h(x) then we have found our lim f(x)

What I'm really confused about is how we set up the squeeze inequality that I see in some textbooks.

Would it be something like this ?

1 =< (5x2y3)/(x2 + y2) <= (I have no idea how you would find an expression on the RHS)
 
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  • #2
It's almost impossible to tell what you mean. If you don't want to use LaTeX or html tags, at least use "^" to indicate a power. I think your "5x2y3" is supposed to mean 5x^2y^3. But you also need parentheses. Do you mean
1)1 +((5x^2y^3)/x^2) + y^2
2) (1+ 5x^2y^3)/ (x^2+ y^2)
3) 1+ 5x^2y^3/(x^2+ y^2)?

I suspect you mean the third but I cannot be certain

I recommend casting into polar coordinates, then getting you "squeeze" by observing that sine and cosine are always between -1 and 1.
 
  • #3
Oh, sorry I did mean the third one. I copied and pasted directly from a pdf file and it messed up the formatting without me realising sorry. Ill try the polar coordinates now thx.
 
  • #4
Okay, So I subbed in x = r cos (alpha) and y = r sin (alpha) and simplified the expression down to

1 + 5r^3(cos(alpha))^2(sin(alpha))^3

My question is, because as the lim r-> 0 then isn't the limit just 1 (which is what I wanted to show) and I wouldn't have to use the squeeze theorem if doing this question in polar coordinates ?
 

FAQ: Squeeze Theorem - Multivarible question

What is the Squeeze Theorem for multivariable functions?

The Squeeze Theorem is a mathematical theorem that states if two functions, f(x,y) and g(x,y), are such that f(x,y) ≤ h(x,y) ≤ g(x,y) for all (x,y) in a neighborhood of (a,b), and if lim(x,y)→(a,b) f(x,y) = L and lim(x,y)→(a,b) g(x,y) = L, then lim(x,y)→(a,b) h(x,y) = L. In other words, if two functions "squeeze" a third function, and all three functions have the same limit at a particular point, then the third function also has the same limit at that point.

How is the Squeeze Theorem used in multivariable calculus?

The Squeeze Theorem is used in multivariable calculus to prove the limit of a multivariable function. It is particularly useful when the limit cannot be evaluated directly using algebraic methods. By finding two functions that bound the function in question and have the same limit, the Squeeze Theorem allows us to conclude that the limit of the function in question is the same as the limit of the bounding functions.

Can the Squeeze Theorem be applied to any multivariable function?

Yes, the Squeeze Theorem can be applied to any multivariable function as long as the functions that bound it have the same limit. However, it is important to note that the Squeeze Theorem only applies to functions that have limits at a particular point.

What is the difference between the Squeeze Theorem and the Sandwich Theorem?

The Squeeze Theorem and the Sandwich Theorem are two names for the same theorem. They both refer to the concept of "squeezing" a function between two other functions to prove its limit. The term "Sandwich Theorem" is often used in introductory calculus courses, while the term "Squeeze Theorem" is more commonly used in higher level mathematics courses.

Can the Squeeze Theorem be extended to more than two functions?

Yes, the Squeeze Theorem can be extended to more than two functions. In fact, the Squeeze Theorem can be extended to any finite number of functions, as long as all of the functions have the same limit at a particular point.

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