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Thread: Limit inside an integral

  1. #1
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    Limit inside an integral

    Which theorem or rule is this:

    $\displaystyle \int_{b}^{c} \lim_{x\to a} f(x) dx = \lim_{x\to a} \int_{b}^{c} f(x) dx$?

    When you can a limit which is inside a integral add before an integral.
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  2. #2
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    Re: Limit inside an integral

    Quote Originally Posted by Nforce View Post
    Which theorem or rule is this:

    $\displaystyle \int_{b}^{c} \lim_{x\to a} f(x) dx = \lim_{x\to a} \int_{b}^{c} f(x) dx$?

    When you can a limit which is inside a integral add before an integral.
    Let's consider

    $\displaystyle \int_0^2 \lim_{x \to 0}x^2dx$ and $\displaystyle \lim_{x \to 0} \int_0^2 x^2dx$

    The LHS and RHS are clearly not equal. I'd say it is rare when the two would be equal. On the LHS, you are taking the integral of a constant. On the RHS, you are taking the limit of a constant.
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    Re: Limit inside an integral

    Quote Originally Posted by Nforce View Post
    Which theorem or rule is this:
    $\displaystyle \int_{b}^{c} \lim_{x\to a} f(x) dx = \lim_{x\to a} \int_{b}^{c} f(x) dx$?
    When you can a limit which is inside a integral add before an integral.
    That a simple example $f(x)=3x^2$
    $\displaystyle \lim_{x\to a} \int_{b}^{c} f(x) dx=ca^3-ba^3$
    $\displaystyle\int_{b}^{c} \lim_{x\to a} f(x) dx = 3ca^2-3ba^2$

    What does that tell us?
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  4. #4
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    Re: Limit inside an integral

    It's not equal.

    But when is equal? Because I saw before in some examples where you can switch the integral and limit. Maybe with infinity?
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    Re: Limit inside an integral

    Quote Originally Posted by Nforce View Post
    It's not equal.

    But when is equal? Because I saw before in some examples where you can switch the integral and limit. Maybe with infinity?
    Do you mean something like this:

    $\displaystyle \lim_{n \to \infty} \int_a^b f_n(x)dx = \int_a^b \lim_{n \to \infty} f_n(x)dx$?

    That is a very different equation from what you wrote. What you wrote, the limits turn the function into a constant. In what I wrote, the limit turns a family of functions into a limit function. Check the Lebesgue Dominated Convergence Theorem and the Monotone Convergence Theorem to start.
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    Re: Limit inside an integral

    Quote Originally Posted by Nforce View Post
    $\displaystyle \int_{b}^{c} \lim_{x\to a} f(x) dx = \lim_{x\to a} \int_{b}^{c} f(x) dx$?
    Note that the left hand side is the integral of a constant, while the right hand side is the limit of a constant.
    Moreover, on the right hand side the $\displaystyle x$ in $\displaystyle \lim_{x \to a}$ is not the same $\displaystyle x$ as the variable of integration $\displaystyle \int_b^c f(x)\,\mathrm dx$
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  7. #7
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    Re: Limit inside an integral

    Quote Originally Posted by Archie View Post
    Note that the left hand side is the integral of a constant, while the right hand side is the limit of a constant.
    Moreover, on the right hand side the $\displaystyle x$ in $\displaystyle \lim_{x \to a}$ is not the same $\displaystyle x$ as the variable of integration $\displaystyle \int_b^c f(x)\,\mathrm dx$
    I pointed that out in post #2.
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    Re: Limit inside an integral

    Quote Originally Posted by SlipEternal View Post
    Do you mean something like this:

    $\displaystyle \lim_{n \to \infty} \int_a^b f_n(x)dx = \int_a^b \lim_{n \to \infty} f_n(x)dx$?

    That is a very different equation from what you wrote. What you wrote, the limits turn the function into a constant. In what I wrote, the limit turns a family of functions into a limit function. Check the Lebesgue Dominated Convergence Theorem and the Monotone Convergence Theorem to start.
    Ok, I read about these theorems, but it applies on the sequences. What do you mean by family of functions? So sequence as a family of function? Can you give a simple example as Plato?

    Thanks.
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  9. #9
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    Re: Limit inside an integral

    Quote Originally Posted by Nforce View Post
    Ok, I read about these theorems, but it applies on the sequences. What do you mean by family of functions? So sequence as a family of function? Can you give a simple example as Plato?

    Thanks.
    $f_n(x) = x+\dfrac{1}{n}$

    $\lim_{n \to \infty} f_n(x) = x$
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    Re: Limit inside an integral

    Ok I understand the example (sort of), but can you be more descriptive?

    So x is an argument of function f, where n is what? So here we mix discrete and analog functions?
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    Re: Limit inside an integral

    Unfortunately, this is not the correct medium for a lecture in real analysis. I'm sure there are plenty of texts, YouTube videos, etc. that could do the topic of convergent sequences (or families) of functions more justice than I could.
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  12. #12
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    Re: Limit inside an integral

    A quick google search found this:
    http://www.personal.psu.edu/auw4/M401-notes1.pdf
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  13. #13
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    Re: Limit inside an integral

    Oh, I think I got it now.

    So for example we have a set A, which contains a sequence of functions or should I say it's a sequence of functions:

    $\displaystyle A = \{f_n, f_m, f_o\}$

    where lets say:

    $\displaystyle f_n = nx$
    $\displaystyle f_m = cos(xm)$
    $\displaystyle f_o = 3o^x$

    In those examples we have just one function in the set.
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  14. #14
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    Re: Limit inside an integral

    Quote Originally Posted by Nforce View Post
    Oh, I think I got it now.

    So for example we have a set A, which contains a sequence of functions or should I say it's a sequence of functions:

    $\displaystyle A = \{f_n, f_m, f_o\}$

    where lets say:

    $\displaystyle f_n = nx$
    $\displaystyle f_m = cos(xm)$
    $\displaystyle f_o = 3o^x$

    In those examples we have just one function in the set.
    So am I correct?
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  15. #15
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    Re: Limit inside an integral

    Quote Originally Posted by Nforce View Post
    Oh, I think I got it now.

    So for example we have a set A, which contains a sequence of functions or should I say it's a sequence of functions:

    $\displaystyle A = \{f_n, f_m, f_o\}$

    where lets say:

    $\displaystyle f_n = nx$
    $\displaystyle f_m = cos(xm)$
    $\displaystyle f_o = 3o^x$

    In those examples we have just one function in the set.
    I don't understand what you wrote at all. You have a set of three functions, $f_n, f_m, f_o$. That is not a sequence of functions (a sequence is a countably infinite set where there is a progression. There is an initial function, and then each function knows the function that comes after it. Example: $f_0, f_1, f_2, \ldots$ is a sequence of functions.

    To clarify this example further, suppose:

    $f_0(x) = 0x = 0$
    $f_1(x) = 1x = x$
    $f_2(x) = 2x$
    $f_3(x) = 3x$
    .
    .
    .
    $f_n(x) = nx$

    That is a sequence of functions.

    There are an infinite number of functions.

    If we fix a point, say $x=c$ for some $c \in \mathbb{R}$, then we look to see does the sequence of numbers, $f_0(c), f_1(c), f_2(c), \ldots$ converge to a single point? It only converges for the point $x=0$. Otherwise, it does not converge.

    Alternately, if we look at the functions:
    $g_1(x) = \dfrac{x}{1}$
    $g_2(x) = \dfrac{x}{2}$
    $g_3(x) = \dfrac{x}{3}$
    .
    .
    .
    $g_n(x) = \dfrac{x}{n}$
    .
    .
    .
    Consider any constant $c \in \mathbb{R}$. We have:

    $\displaystyle \lim_{n \to \infty} g_n(c) = \lim_{n \to \infty} \dfrac{c}{n} = c\lim_{n \to \infty} \dfrac{1}{n} = c(0) = 0$

    This converges point-wise to the zero function. So, we can say: $\displaystyle \lim_{n \to \infty} g_n(x) = g(x) = 0$.

    Consider the sequence of functions:
    $h_1(x) = x+\dfrac{1}{1}$
    $h_2(x) = x+\dfrac{1}{2}$
    $h_3(x) = x+\dfrac{1}{3}$
    .
    .
    .
    $h_n(x) = x + \dfrac{1}{n}$
    .
    .
    .

    This sequence of function converges to the function $h(x) = x$.
    Last edited by SlipEternal; Oct 17th 2017 at 12:35 PM.
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