Limit of f(x,y) as x^2+y^2→∞: Calculate & Analyze

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In summary, the conversation discusses determining the limit of the function f(x,y)=\frac{\ln(1+x^2y^2)}{x^4+y^4} as x^2+y^2 approaches infinity. The speaker suggests using L'Hopital's rule and simplifying the expression in polar form, eventually concluding that the limit is equal to 0. They also mention the use of an upper bound for the natural logarithm function and discuss whether it is allowed to use r^2->\infty instead of r->\infty when calculating limits.
  • #1
Petrus
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\(\displaystyle f(x,y)=\frac{\ln(1+x^2y^2)}{x^4+y^4}\)
decide if it got a limit if \(\displaystyle x^2+y^2=\infty\). if so calculate it.

well I go to polar form and we got
\(\displaystyle \lim_{r^2->\infty}\frac{\ln(1+r^2\cos^2(\theta)r^2\sin^2( \theta))}{r^4\cos^2(\theta)^4+r^4\sin^4(\theta)}\)
we see both approach to limit but the bottom will go a lot faster so it will be equal to zero, is this wrong to say like this?

Regards,
\(\displaystyle |\pi\rangle\)
 
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  • #2
Petrus said:
\(\displaystyle f(x,y)=\frac{\ln(1+x^2y^2)}{x^4+y^4}\)
decide if it got a limit if \(\displaystyle x^2+y^2=\infty\). if so calculate it.

well I go to polar form and we got
\(\displaystyle \lim_{r^2->\infty}\frac{\ln(1+r^2\cos^2(\theta)r^2\sin^2( \theta))}{r^4\cos^2(\theta)^4+r^4\sin^4(\theta)}\)
we see both approach to limit but the bottom will go a lot faster so it will be equal to zero, is this wrong to say like this?

Regards,
\(\displaystyle |\pi\rangle\)

Hi Petrus, :)

I would use the L'Hopital's rule over the variable \(r\).
 
  • #3
Sudharaka said:
Hi Petrus, :)

I would use the L'Hopital's rule over the variable \(r\).
Thanks! Now I see without finish the L'Hopital's rule as you need to do it 4 times :P we will end with a \(\displaystyle 24\cos^2(\theta)sin^2(\theta)\) at top and we will keep have some r at bottom which will make bottom to \(\displaystyle \infty\) so it will equal to 0!:) I got one question that I would like to know.

is it allowed to say \(\displaystyle r^2->\infty\) is same as \(\displaystyle r->\infty\) when you calculate limits?

Regards,
\(\displaystyle |\pi\rangle\)
 
  • #4
Petrus said:
Thanks! Now I see without finish the L'Hopital's rule as you need to do it 4 times :P we will end with a \(\displaystyle 24\cos^2(\theta)sin^2(\theta)\) at top and we will keep have some r at bottom which will make bottom to \(\displaystyle \infty\) so it will equal to 0!:) I got one question that I would like to know.

Alternatively, you can use the upper bound $\ln(1+u) \le \sqrt u$.

Btw, do you need 4 applications of l'Hospital's rule? It seems I only need 1.

is it allowed to say \(\displaystyle r^2->\infty\) is same as \(\displaystyle r->\infty\) when you calculate limits?

It's unusual, but I don't think it's wrong.
It fits into the definition of a limit.
 
  • #5
I like Serena said:
Alternatively, you can use the upper estimate $\ln(1+u) \le \sqrt u$.

Btw, do you need 4 applications of l'Hospital's rule? It seems I only need 1.
It's unusual, but I don't think it's wrong.
It fits into the definition of a limit.
you are correct! we will be able to divide \(\displaystyle 4r^3\) on top and bottom! Thanks!

Regards,
\(\displaystyle |\pi\rangle\)
 

FAQ: Limit of f(x,y) as x^2+y^2→∞: Calculate & Analyze

1. What is the concept of a limit in mathematics?

A limit is a fundamental concept in mathematics that describes the behavior of a function or sequence as its input values approach a certain value or infinity. It represents the value that a function or sequence is approaching, rather than the actual value at that point.

2. How do you calculate the limit of a function?

The limit of a function can be calculated by evaluating the function at values closer and closer to the desired input value. This can be done numerically or algebraically, using techniques such as factoring, rationalizing, or L'Hopital's rule.

3. What does it mean for a limit to approach infinity?

When a limit approaches infinity, it means that the input values of the function are becoming increasingly large. In other words, as the input values increase without bound, the output values of the function also increase without bound.

4. How do you analyze the limit of a function as x^2+y^2 approaches infinity?

To analyze the limit of a function as x^2+y^2 approaches infinity, you can first consider the behavior of the function at various points on the graph. Then, you can examine the behavior as the input values approach infinity, and use techniques such as factoring, rationalizing, or L'Hopital's rule to evaluate the limit.

5. What can the limit of a function tell us about its behavior?

The limit of a function can provide information about the overall behavior of the function, such as the existence of asymptotes or the presence of discontinuities. It can also help determine the end behavior of the function as the input values approach infinity or negative infinity.

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