How to Prove a Multivariable Function Converges to a Specific Value?

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In summary, to prove that the given function converges to ln(2) in the case of x=0 and y=0, one can use the above properties and the fact that the function can be written as a product of two single-variable functions.
  • #1
PsychonautQQ
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Homework Statement


How do I prove the function ((2^x)-1)sin(y))/(xy) converges to ln(2) in the case of x=0 and y=0..?


Homework Equations





The Attempt at a Solution


Yeah I'm trying to figure out where to start honestly... I know I'm not suppose to post unless I have a real attempt at the solution.. I could BS something but anyone want to point me in the right direction? Squeeze Theorem? Is there some key property I should be aware of?
 
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  • #2
PsychonautQQ said:

Homework Statement


How do I prove the function ((2^x)-1)sin(y))/(xy) converges to ln(2) in the case of x=0 and y=0..?


Homework Equations





The Attempt at a Solution


Yeah I'm trying to figure out where to start honestly... I know I'm not suppose to post unless I have a real attempt at the solution.. I could BS something but anyone want to point me in the right direction? Squeeze Theorem? Is there some key property I should be aware of?

Why do you think the answer is ##\ln 2##? Try some special limits to get a feel for it. What if ##y\rightarrow 0## before ##x##, for example.
 
  • #3
the answer is ln(2), the question says that, we just have to PROVE it is.
 
  • #4
Yes, the answer is indeed ln(2) I used mclaurin series to prove it.
 
  • #5
PsychonautQQ said:

Homework Statement


How do I prove the function ((2^x)-1)sin(y))/(xy) converges to ln(2) in the case of x=0 and y=0..?


Homework Equations





The Attempt at a Solution


Yeah I'm trying to figure out where to start honestly... I know I'm not suppose to post unless I have a real attempt at the solution.. I could BS something but anyone want to point me in the right direction? Squeeze Theorem? Is there some key property I should be aware of?

Your function is a function of x only times a function of y only:
[tex]
\frac{(2^x - 1)\sin y}{xy} = \frac{2^x - 1}{x} \frac{\sin y}{y}
[/tex]

One can show that if [itex]\lim_{x \to 0} f(x)[/itex] and [itex]\lim_{y \to 0} g(y)[/itex] exist then
[tex]
\lim_{(x,y) \to 0} f(x)g(y) = \left(\lim_{x \to 0} f(x) \right)\left( \lim_{y \to 0} g(y) \right)
[/tex]

Also remember that
[tex]
\lim_{x \to 0} \frac{f(x) - f(0)}x = f'(0)
[/tex]
assuming the limit exists.
 

FAQ: How to Prove a Multivariable Function Converges to a Specific Value?

1. What is function convergence?

Function convergence is the process of determining whether a function approaches a specific value or not as its input variable approaches a certain value. It is an important concept in mathematical analysis and is often used to determine the behavior of functions in various scenarios.

2. How do you prove that a function converges?

To prove that a function converges, you need to show that for any arbitrarily small value, there exists a corresponding value of the input variable where the function's output will be within that small value. This can be done using various mathematical techniques such as the epsilon-delta definition or the Cauchy criterion.

3. Can a function converge to multiple values?

No, a function can only converge to a single value. This is because the definition of convergence requires the function to approach a specific value as its input variable approaches a certain value. If a function converges to multiple values, it would not have a well-defined limit.

4. What is the difference between pointwise and uniform convergence?

Pointwise convergence is when a function converges at each individual point in its domain. Uniform convergence, on the other hand, is when a function converges at every point in its domain simultaneously. In other words, for uniform convergence, the rate of convergence is the same at every point in the domain.

5. Why is proving function convergence important?

Proving function convergence is important because it allows us to understand the behavior of a function and make predictions about its outputs. It is also a fundamental concept in calculus and real analysis, and is used in many applications such as optimization, physics, and engineering.

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