Solving Limits and Convergence: Doubts and Calculations Explained

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In summary, Paul had a doubt about whether his approach to determining the existence of a limit was correct, as it seemed too easy. He also had a second doubt about the convergence of an integral. Other users provided feedback and confirmed that his approach to the second problem was correct. Paul thanked everyone for their help.
  • #1
pbialos
Hi, Today i had a test, and i was wondering if what i did is correct:

I had to tell if the [tex]:lim_{(x,y) \rightarrow (0,0)} \frac {x*y} {|x|+|y|}[/tex] exists. What i did was to say [tex]lim_{(x,y) \rightarrow (0,0)} \frac {x*y} {|x|+|y|}=lim_{(x,y) \rightarrow (0,0)} \frac {x} {|x|+|y|}*y=0[/tex] because it is bounded multiplied by y that tends to 0.
Is what i did correct?Something tells me it is not because it was too easy.

The second doubt i have is about the convergence of the Integral:
[tex]\int_{0}^{\infty}\frac {sin(x)} {cos(x)+x^2}[/tex]
My doubt came since the integrand of the series is not always positive. Can i just calculate the convergence of:
[tex]\int_{0}^{\infty}|\frac {sin(x)} {cos(x)+x^2}|[/tex] and if it converges, then the original integral also converges?
I would check the convergence of the second integral dividing it between 0-1 and 1-infinity. The integral on the first interval(between 0 and 1) would converge because the integrand is bounded and continuous for x between 0 and 1, and the integral of the second interval(between 1 and infinity) would also converge by comparison with [tex]\int_{1}^{\infty}\frac {1} {x^2-1}[/tex].

I would really appreciate any help.
Regards, Paul.
 
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  • #2
I'm rusty on stuff like convergence, but what you did with the second problem seems right. However, are you sure that [itex]\int _1 ^{\infty }\frac{1}{x^2 - 1}\, dx[/itex] converges?
 
Last edited:
  • #3
AKG said:
On the other hand, approach along y = 0, you get:

[tex]\lim _{(x,0) \to (0,0)} \frac{0x}{|x| + |0|} = \lim _{x \to 0}\frac{0}{|x|} = \infty[/tex]

Since the numerator is zero along that path, the limit will clearly also be zero along that path.

pbialos, I think you have the right idea. It is indeed not difficult, but the phrasing was a bit vague. It's not clear to me what you mean by "because it is bounded multiplied by y that tends to 0."

Try something like:

[tex]\left|\frac {xy} {|x|+|y|}\right|=|y| \frac{|x|}{|x|+|y|} \leq |y|[/tex]
because [itex]|x|/(|x|+|y|) \leq 1[/itex]. Since |y| approaches zero as (x,y) approaches (0,0), the limit is zero.
 
  • #4
Galileo said:
Since the numerator is zero along that path, the limit will clearly also be zero along that path.
Yeah, I don't know what I was thinking... edited.
 
  • #5
thank you

Many thanks for your responses. Yes, what Galileo said was my idea, although i expressed my self terribly bad.

Regards, Paul.
 

1. What is the definition of a limit?

A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. It represents the value that the function is approaching, or "approaches asymptotically," as the input gets closer and closer to the specified value.

2. How do you solve a limit?

To solve a limit, you can use algebraic techniques, such as factoring and simplifying, to manipulate the function into a form that is easier to evaluate. You can also use graphical methods, such as plotting the function and observing its behavior near the specified value. Additionally, you can use analytical methods, such as the squeeze theorem or L'Hopital's rule, to evaluate the limit.

3. What is the difference between a one-sided and two-sided limit?

A one-sided limit only considers the behavior of the function as the input approaches the specified value from one direction (either the left or the right). A two-sided limit, on the other hand, considers the behavior of the function as the input approaches the specified value from both directions.

4. How do you determine whether a limit exists or not?

A limit exists if the values of the function approach a single value as the input approaches the specified value. This means that the left-sided and right-sided limits must be equal. If the two-sided limit does not exist, then the overall limit does not exist.

5. What is convergence and how does it relate to limits?

Convergence is a property of a sequence or series that indicates whether the terms of the sequence or series approach a finite value as the number of terms increases. In the context of limits, convergence refers to the behavior of the function as the input approaches a specific value. A limit is said to converge if the function approaches a single value as the input gets closer and closer to the specified value.

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