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How to treat limits such as these properly? Watch

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    Say we have some limit.

    \displaystyle \lim_{x\rightarrow \infty }(\sqrt{x+10}-\sqrt x~)

    and I need to evaluate this limit.

    What is the best way to do this obviously showing my working I know that sqrt(x+10) and sqrt(x) both tend to infinity and so the limit of function should be the limits of both parts taken away which is "infinity minus infinity" which is 0. I'm not sure whether this is a valid method and if it isn't what should I be doing?

    Any help?

    Thanks
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    Also with this as well:

    I see the limit is 13/23 since sinx~x for small x so the limit is 13/23 but I don't know how to write down a proper evaluation.
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    (Original post by alex2100x)
    Also with this as well:

    I see the limit is 13/23 since sinx~x for small x so the limit is 13/23 but I don't know how to write down a proper evaluation.
    Consider using the Taylor expansion of sin(ax), and look at how the terms shrink as x approaches zero. Then it should hopefully become clear how to evaluate the limit.

    For the first problem, you have to consider the fact that as x approaches infinity, the square root of (x+10) is approximately the same as the square root of x. Because of that, the limit as x goes to infinity is zero. Also, note that you can take the limit of the two functions as x approaches zero independently, and then subtract the results. In terms of doing it rigorously, you could perhaps argue that:

    lim(sqrt(x+10) - sqrt(x)) = lim(sqrt(x+10)) - lim(sqrt(x)) = lim(sqrt(x)) - lim(sqrt(x)) = 0

    But I'm not sure how a mathematician would view that. It'd probably be frowned upon, and there's probably a better way to do it, but it'd convince me.

    What you cannot do however is simply state that because both limits are infinity, one minus the other must be zero. That isn't true - infinity minus infinity is an indeterminate form, since not all infinities are the same.

    For example, if you took the sum of every integer, which is obviously infinite, and subtracted from it the sum of every odd integer, which is also infinite, your answer would not be zero.
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    (Original post by alex2100x)
    Say we have some limit.

    \displaystyle \lim_{x\rightarrow \infty }(\sqrt{x+10}-\sqrt x~)

    and I need to evaluate this limit.

    What is the best way to do this obviously showing my working I know that sqrt(x+10) and sqrt(x) both tend to infinity and so the limit of function should be the limits of both parts taken away which is "infinity minus infinity" which is 0. I'm not sure whether this is a valid method and if it isn't what should I be doing?

    Any help?

    Thanks
    Also, try multiplying the numerator and denominator of the argument of the limit by its conjugate, and see where that gets you. This is the best method I can come up with having put some thought into it!
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    (Original post by arcturus7)
    Also, try multiplying the numerator and denominator of the argument of the limit by its conjugate, and see where that gets you. This is the best method I can come up with having put some thought into it!
    Thanks, I get the idea I just want to make sure my work is "acceptable".
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    (Original post by alex2100x)
    Thanks, I get the idea I just want to make sure my work is "acceptable".
    Well infinity minus infinity = 0 is definitely not true, and is probably unacceptable.

    My first suggestion is probably slightly shaky, because although it tries to subtract the same limit from itself, I don't know if that is rigorous enough.

    The multiplication thing is guaranteed to work because you end up with 10/infinity which is definitely zero.
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    You can apply l'hospitals rule for the quotient function.


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    (Original post by alex2100x)
    Also with this as well:

    I see the limit is 13/23 since sinx~x for small x so the limit is 13/23 but I don't know how to write down a proper evaluation.
    You know that if the limit of \displaystyle \lim_{x \rightarrow a}f exists, then you know the limit of \displaystyle \lim_{x \rightarrow a}\dfrac{1}{f} also exists.

    So \displaystyle \dfrac{sin(13x)}{sin(23x)} = \dfrac{sin(13x)}{13x}\dfrac{23x}  {sin(23x)}\dfrac{13x}{23x}. So the limit must be 13/23.


    (Original post by arcturus7)
    Well infinity minus infinity = 0 is definitely not true, and is probably unacceptable.

    My first suggestion is probably slightly shaky, because although it tries to subtract the same limit from itself, I don't know if that is rigorous enough.

    The multiplication thing is guaranteed to work because you end up with 10/infinity which is definitely zero.
    But infinity - infinity can equal zero. i.e \displaystyle \lim_{x \rightarrow \infty} x - x.

    OP can just do standard epsilon-delta symbol shunting.

    |(\sqrt{x + 10} - \sqrt{x}) - 0| = \sqrt{x + 10} - \sqrt{x} < \epsilon.

    Then rearrange to get \displaystyle \left(\dfrac{10 - \epsilon^2}{2\epsilon}\right)^2 < x. Now you can cook up the usual epsilon-delta proof.
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    (Original post by 0x2a)
    You know that if the limit of \displaystyle \lim_{x \rightarrow a}f exists, then you know the limit of \displaystyle \lim_{x \rightarrow a}\dfrac{1}{f} also exists.

    So \displaystyle \dfrac{sin(13x)}{sin(23x)} = \dfrac{sin(13x)}{13x}\dfrac{23x}  {sin(23x)}\dfrac{13x}{23x}. So the limit must be 13/23.



    But infinity - infinity can equal zero. i.e \displaystyle \lim_{x \rightarrow \infty} x - x.

    OP can just do standard epsilon-delta symbol shunting.

    |(\sqrt{x + 10} - \sqrt{x}) - 0| = \sqrt{x + 10} - \sqrt{x} < \epsilon.

    Then rearrange to get \displaystyle \left(\dfrac{10 - \epsilon^2}{2\epsilon}\right)^2 < x. Now you can cook up the usual epsilon-delta proof.
    Perhaps I was unclear, yes infinity - infinity can be zero, but it isn't always true.

    I was saying that you cannot make the general statement that inf-inf=0, because you could easily cook up a limit which is infinity - infinity but =/= 0.
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    (Original post by alex2100x)
    Say we have some limit.

    \displaystyle \lim_{x\rightarrow \infty }(\sqrt{x+10}-\sqrt x~)

    and I need to evaluate this limit.

    What is the best way to do this obviously showing my working I know that sqrt(x+10) and sqrt(x) both tend to infinity and so the limit of function should be the limits of both parts taken away which is "infinity minus infinity" which is 0. I'm not sure whether this is a valid method and if it isn't what should I be doing?

    Any help?

    Thanks
    For this one, the argument \displaystyle \lim_{x \to \infty} (\sqrt{x+10}) = \lim_{x \to \infty}\sqrt{x} is not rigorous, because "limit of the sum is the sum of the limits" only is allowed when the limits are finite.

    However, let L = \lim_{x \to \infty} (\sqrt{x+10} - \sqrt{x}).

    "Rationalise the denominator" to get \displaystyle L = \lim_{x \to \infty} \frac{10}{\sqrt{x+10}+\sqrt{x}}. (We've multiplied by the fraction \dfrac{\sqrt{x+10} + \sqrt{x}}{\sqrt{x+10} + \sqrt{x}}.) That's obviously got limit 0.
 
 
 
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