Integrals & Limits: Intuitive Understanding of Convergence

In summary: I'll just plant this "rule" into my brain for now and then see if I can understand it better conceptually once we do learn more about limits. But for now, is possible for you to draw a picture to illustrate why the limit has to equal to 0 and not any other constants?
  • #1
MermaidWonders
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Suppose that $\int_{-\infty}^{\infty} f(x)\,dx$ converges. Then $\lim_{{x}\to{-\infty}}f(x) = \lim_{{x}\to{\infty}}f(x)$. Why is it true? I have some trouble understanding this intuitively.
 
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  • #2
MermaidWonders said:
Suppose that $\int_{-\infty}^{\infty} f(x)\,dx$ converges. Then $\lim_{{x}\to{-\infty}}f(x) = \lim_{{x}\to{\infty}}f(x)$. Why is it true? I have some trouble understanding this intuitively.

Hey MermaidWonders!

Let's start with $\int_0^{\infty} f(x)\,dx$, which must also converge.
Then $\lim\limits_{{x}\to{\infty}}f(x) = 0$ isn't it? (Wondering)
Because otherwise it wouldn't converge.
 
  • #3
Oh, OK, but why can't the limit equal to any other number other than 0 for it to converge?
 
  • #4
MermaidWonders said:
Oh, OK, but why can't the limit equal to any other number other than 0 for it to converge?

Suppose $\lim\limits_{x\to\infty} f(x)=c > 0$.
Then for $\varepsilon = \frac c 2 > 0$ there must be some $a$ such that for every $x>a$ we have that $f(x)>c-\varepsilon=\frac c2$.
That's part of what 'limit' means.
But then $\int_a^\infty f(x)\,dx > \int_a^\infty \frac c2\,dx \to \infty$ meaning it does not converge, which is a contradiction.
 
  • #5
What's $\varepsilon$ and $a$? Do these variables pertain to the precise definition of a limit? If so, this is something we haven't touched on yet, so would you mind explaining their significance with respect to this question? Also, is there another way to approach this question using the regular/basic definition of limits? Thanks!
 
  • #6
MermaidWonders said:
What's $\varepsilon$ and $a$? Do these variables pertain to the precise definition of a limit? If so, this is something we haven't touched on yet, so would you mind explaining their significance with respect to this question? Also, is there another way to approach this question using the regular/basic definition of limits? Thanks!

What is the regular/basic definition of limits that you are familiar with?
The generic definition is the so-called $\varepsilon$-$\delta$ definition, which may come later for you.
 
  • #7
Oh, oops. By "regular/basic definition" of limits, I meant like limits of the basic form $\lim_{{x}\to{a}}f(x)$... And yeah, the generic definition of the limit you have up there is unfamiliar to me too, but I do suppose we'll learn it later.
 
  • #8
MermaidWonders said:
Oh, oops. By "regular/basic definition" of limits, I meant like limits of the basic form $\lim_{{x}\to{a}}f(x)$... And yeah, the generic definition of the limit you have up there is unfamiliar to me too, but I do suppose we'll learn it later.

Erm... then I'm not quite sure what I can use or not.
Can you perhaps use the fact that a series $x_1 + x_2 + ...$ can only converge if $x_n \to 0$?
Or that an improper integral like $\int_0^\infty f(x)\,dx$ can only converge if $\lim_{x\to\infty} f(x) = 0$?
 
  • #9
Yeah, sure. :) I'll just plant this "rule" into my brain for now and then see if I can understand it better conceptually once we do learn more about limits. But for now, is possible for you to draw a picture to illustrate why the limit has to equal to 0 and not any other constants? That's what I'm still having trouble with.
 
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  • #10
MermaidWonders said:
Yeah, sure. I'll just plant this "rule" into my brain for now and then see if I can understand it conceptually once we do learn more about limits. But for now, is possible for you to draw a picture to illustrate why the limit has to equal to 0 and not any other constants? That's what I'm still having trouble with.

Conceptually an integral corresponds to the area under a graph.
If that graph extends to infinity, the height of the graph must approach zero, since otherwise that area tends to infinity.
 
  • #11
Oh, so that's it? Then does it mean that $\lim_{{x}\to{-\infty}}f(x)$ would also have to approach 0 in order for the integral to converge during the entire stretch of the interval from $-\infty$ to $\infty$?
 
  • #12
MermaidWonders said:
Oh, OK, but why can't the limit equal to any other number other than 0 for it to converge?

Another way of looking at this is to visualize $\int_0^\infty c\,dx$, where $c>0$. Now make a drawing, including the $x$-axis, of what this integral looks like in terms of an area between $c$ and the $x$-axis. What do you notice?
 
  • #13
MermaidWonders said:
Oh, so that's it? Then does it mean that $\lim_{{x}\to{-\infty}}f(x)$ would also have to approach 0 in order for the integral to converge during the entire stretch of the interval from $-\infty$ to $\infty$?

Yep. (Nod)
 
  • #14
greg1313 said:
Another way of looking at this is to visualize $\int_0^\infty c\,dx$, where $c>0$. Now make a drawing, including the $x$-axis, of what this integral looks like in terms of an area between $c$ and the $x$-axis. What do you notice?
It diverges? Since the area under the "curve" tends towards infinity as the upper limit of integration is $\infty$.

- - - Updated - - -

Thanks for all the help, guys! :) 👍👍
 
  • #15
MermaidWonders said:
It diverges? Since the area under the "curve" tends towards infinity as the upper limit of integration is $\infty$.

Basically, that's it.
 

Related to Integrals & Limits: Intuitive Understanding of Convergence

1. What is the purpose of understanding convergence in integrals and limits?

Understanding convergence in integrals and limits is important for accurately calculating the values of integrals and limits, which are essential in many areas of mathematics and science. Additionally, understanding convergence helps to ensure the validity and accuracy of mathematical models and equations.

2. How can I intuitively understand convergence in integrals and limits?

To intuitively understand convergence, it is helpful to think of it as the process of approaching a certain value or limit as the input or variable gets closer to a specific value. In integrals, convergence is achieved when the upper and lower limits of integration approach a specific value and the resulting area under the curve approaches a finite value.

3. What are some common tests for convergence in integrals and limits?

Some common tests for convergence in integrals include the comparison test, the limit comparison test, the ratio test, and the integral test. In limits, common tests for convergence include the squeeze theorem, the limit comparison test, and the ratio test.

4. What are some common misconceptions about convergence in integrals and limits?

One common misconception is that a series or integral must approach a specific value in order to converge, when in reality, it may converge to a range of values. Another misconception is that a series or integral must approach a certain value at a certain rate, when in reality, it may approach the value at a slower or faster rate depending on the specific convergence test used.

5. Why is it important to check for convergence in mathematical calculations?

Checking for convergence is important because it ensures the accuracy and validity of mathematical calculations. Inaccurate calculations can lead to incorrect conclusions and flawed mathematical models, which can have significant consequences in various fields such as physics, engineering, and economics.

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