Integrals & Limits: Intuitive Understanding of Convergence

[MermaidWonders]

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.

 

[I like Serena]

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.

 

[MermaidWonders]

Oh, OK, but why can't the limit equal to any other number other than 0 for it to converge?

 

[I like Serena]

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.

 

[MermaidWonders]

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!

 

[I like Serena]

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.

 

[MermaidWonders]

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.

 

[I like Serena]

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 $\lim_{x\to\infty} f(x) = 0$?

 

[MermaidWonders]

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.

 

[I like Serena]

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.

 

[MermaidWonders]

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 $\infty$?

 

[Greg]

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?

 

[I like Serena]

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)

 

[MermaidWonders]

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 $\infty$.

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Thanks for all the help, guys! :)

 

[Greg]

It diverges? Since the area under the "curve" tends towards infinity as the upper limit of integration is $\infty$.

Basically, that's it.