Sandwich Theorem proof lim (x,y) -> (0,0) (4x^3*y^4)/(3y^4+5x^8)

In summary, the conversation discusses finding the limit of a function involving x and y as they approach (0,0), and various strategies for proving that the limit is equal to 0. The final solution involves using absolute value signs around the entire fraction and taking the limit as x approaches 0.
  • #1
Phyrrus
21
0

Homework Statement



lim (x,y) -> (0,0) (4x^3*y^4)/(3y^4+5x^8)

When y=kx -> lim x->0 = 0
The limit definitely is zero, but I can't prove it.

The Attempt at a Solution



? < lim (x,y) -> (0,0) (4x^3*y^4)/(3y^4+5x^8) < lim (x,y) -> (0,0) (4|x^3|*y^4+(15/4)|x^3|x^8)/(3y^4+5x^8) = (3/4)|x^3|

How do you find the lower limit of such a proof?
Can it be said that lim (x,y) -> (0,0) (4x^3*y^4)/(3y^4+5x^8) is equal to lim (x,y) -> (0,0) (|4x^3|*y^4)/(3y^4+5x^8) and therefore find the 'sandwich' of the new function?
 
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  • #2
Hint: Your denominator is non-negative. If you drop the ##5x^8## from the denominator, the fraction will be larger in absolute value than it is now.
 
  • #3
Thanks, that is quite a simple solution. However, what can I use for the smaller valued function? Seeing as there is an x^3 in the numerator, I can't use 0 for the lower limit.
 
  • #4
Oh wait, I just need to put in a negative sign for the lower limit... it's late

Cheers mate
 
  • #5
Phyrrus said:
Thanks, that is quite a simple solution. However, what can I use for the smaller valued function? Seeing as there is an x^3 in the numerator, I can't use 0 for the lower limit.

You can if you put absolute value signs around the whole fraction. Remember if the absolute value of something goes to zero, then the something does.
 

Related to Sandwich Theorem proof lim (x,y) -> (0,0) (4x^3*y^4)/(3y^4+5x^8)

1. What is the Sandwich Theorem?

The Sandwich Theorem, also known as the Squeeze Theorem, is a mathematical principle used to prove the limit of a function by comparing it between two other functions that have the same limit.

2. How is the Sandwich Theorem applied to this specific proof?

In this proof, we are using the Sandwich Theorem to evaluate the limit of the function (4x^3*y^4)/(3y^4+5x^8) as (x,y) approaches (0,0). By finding two other functions that are always greater than or less than this function and have the same limit, we can use the Sandwich Theorem to prove the limit of the original function.

3. What are the two functions used in this proof?

The two functions used in this proof are g(x,y) = 4x^3*y^4 and h(x,y) = 3y^4+5x^8. These two functions are chosen because they are always greater than or less than the original function and have the same limit as (x,y) approaches (0,0).

4. How do we determine the limit of the original function using the Sandwich Theorem?

First, we evaluate the limit of the two functions g(x,y) and h(x,y) as (x,y) approaches (0,0). If the limits of these two functions are equal, then we know that the limit of the original function must also be equal to that value. However, if the limits of g(x,y) and h(x,y) are different, then the Sandwich Theorem cannot be used to prove the limit of the original function.

5. Why is the Sandwich Theorem considered a powerful tool in mathematical proofs?

The Sandwich Theorem is considered a powerful tool in mathematical proofs because it allows us to prove the limit of a function without using the traditional methods of direct substitution or algebraic manipulation. It also provides a more intuitive understanding of limits by comparing the function to other known functions.

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