Proof of Calculus III: Proving Limit as (x,y) -> (0,0) = 0

In summary, if the ratio of (a/c) + (b/d) is greater than 1, then the limit as (x,y) approaches (0,0) of f(x,y) exists and equals zero. This is shown by proving that the limit of |f(x,y)| is zero, which is equivalent to showing that the limit of 1/f(x,y) is infinity.
  • #1
Mona1990
13
0
Hi!
I was wondering if someone could give me a couple hints on how to tackle the following proof!

Let f(x,y)= [ (lxl ^a)(lyl^b) ]/ [(lxl^c) + lyl^d] where a,b,c,d are positive numbers.
prove that if (a/c) + (b/d) > 1
then limit as (x,y) -> (0,0) of f(x,y) exists and equals zero.

thanks!
 
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  • #2
The limit as (x,y) goes to (0,0) of f(x,y) is zero if and only if the limit (x,y) -> (0,0) of |f(x,y)| =0

Since |f(x,y)| is always positive, this is equivalent to saying

[tex] \lim_{(x,y) \rightarrow (0,0)} \frac{1}{|f(x,y)|} = \infty[/tex]

This works for general f(x,y). In this case, we know f(x,y) > 0 if (x,y) is not equal to (0,0) so you just need to show[tex] \lim_{(x,y) \rightarrow (0,0)} \frac{1}{f(x,y)} = \infty[/tex]

And this is easier since you can split up the numerator and start comparing a to c and b to d
 
  • #3
Hey!
thanks a lot :) makes sense now!
 

FAQ: Proof of Calculus III: Proving Limit as (x,y) -> (0,0) = 0

What is Calculus III?

Calculus III is the third level of calculus, also known as multivariable calculus. It focuses on the study of functions with multiple variables and their derivatives, integrals, and limits.

What is the limit as (x,y) approaches (0,0)?

The limit as (x,y) approaches (0,0) is the value that a function approaches as the input (x,y) gets closer and closer to (0,0). It helps determine the behavior of a function at a specific point.

Why is proving the limit as (x,y) approaches (0,0) important?

Proving the limit as (x,y) approaches (0,0) is important because it helps us understand the behavior of a function at a specific point. It also allows us to make predictions and solve problems involving multivariable functions.

How is the limit as (x,y) approaches (0,0) proven?

The limit as (x,y) approaches (0,0) is proven using the epsilon-delta definition. This involves showing that for any small value of epsilon, there exists a corresponding value of delta such that the distance between the input (x,y) and (0,0) is less than delta, and the output of the function is within epsilon of the limit value.

What are some real-life applications of proving the limit as (x,y) approaches (0,0)?

The limit as (x,y) approaches (0,0) has applications in physics, engineering, economics, and many other fields. It can be used to analyze the motion of objects, optimize processes, and model complex systems. For example, in physics, it can help determine the velocity and acceleration of an object at a specific point, while in economics it can help predict the demand for a product at a certain price point.

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