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Thread: Prove lim(x->3) x^2 = 9 by definition

  1. #1
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    Arrow Prove lim(x->3) x^2 = 9 by definition

    Hello!

    I know how to solve this task but I have certain things that I do not understand.

    Solution.

    $\displaystyle [|x^2-9|< \epsilon $ if $\displaystyle 0 < |x-3| < \delta$

    Because |x-3| occurs on the right side of this "if statement", it will be helpful to factor the left side to introduce a factor of |x-3|.

    $\displaystyle |x+3||x-3|< \epsilon$ if $\displaystyle 0<|x-3|<\delta$

    Using the triangle inequality:

    $\displaystyle |x+3|=|(x-3)+6| \leq |x-3| + 6$

    By multiplying the statement above with |x-3| (which is positive) we got:

    $\displaystyle |x+3||x-3| \leq (|x-3| +6)|x-3| $

    Now clearly $\displaystyle 6+|x-3|< 6 + \delta$ and $\displaystyle |x-3|<\delta$

    By multiplying both inequalities (I am not sure if this step is valid, please confirm me) I got:

    $\displaystyle (6+|x-3|)|x-3|< (6 + \delta)\delta$

    We can conclude that:

    $\displaystyle |x+3||x-3| < (6 + \delta)\delta$ if $\displaystyle 0 < |x-3| < \delta$

    Now isn't $\displaystyle \epsilon = (6 + \delta)\delta$ ??

    Why in my book they choose $\displaystyle (6 + \delta)\delta \leq \epsilon$ ?

    And why they choose $\displaystyle \delta \geq 1 $ and with that restriction
    $\displaystyle (6 + \delta)\delta \leq 7\delta$ and $\displaystyle 7\delta \leq \epsilon$.

    Finally I don't understand why $\displaystyle \delta = min(e/7,1)$.

    Could somebody possibly explain?

    Thanks a lot.
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  2. #2
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    Quote Originally Posted by p0oint View Post
    Hello!

    I know how to solve this task but I have certain things that I do not understand.

    Solution.

    $\displaystyle [|x^2-9|< \epsilon $ if $\displaystyle 0 < |x-3| < \delta$

    Because |x-3| occurs on the right side of this "if statement", it will be helpful to factor the left side to introduce a factor of |x-3|.

    $\displaystyle |x+3||x-3|< \epsilon$ if $\displaystyle 0<|x-3|<\delta$

    Using the triangle inequality:

    $\displaystyle |x+3|=|(x-3)+6| \leq |x-3| + 6$

    By multiplying the statement above with |x-3| (which is positive) we got:

    $\displaystyle |x+3||x-3| \leq (|x-3| +6)|x-3| $

    Now clearly $\displaystyle 6+|x-3|< 6 + \delta$ and $\displaystyle |x-3|<\delta$

    By multiplying both inequalities (I am not sure if this step is valid, please confirm me) I got:

    $\displaystyle (6+|x-3|)|x-3|< (6 + \delta)\delta$

    We can conclude that:

    $\displaystyle |x+3||x-3| < (6 + \delta)\delta$ if $\displaystyle 0 < |x-3| < \delta$

    Now isn't $\displaystyle \epsilon = (6 + \delta)\delta$ ??

    Why in my book they choose $\displaystyle (6 + \delta)\delta \leq \epsilon$ ?

    And why they choose $\displaystyle \delta \geq 1 $ and with that restriction
    $\displaystyle (6 + \delta)\delta \leq 7\delta$ and $\displaystyle 7\delta \leq \epsilon$.

    Finally I don't understand why $\displaystyle \delta = min(e/7,1)$.

    Could somebody possibly explain?

    Thanks a lot.
    If you reorganized your work, you will find it as follows:

    $\displaystyle |x^2-9|=|x+3||x-3|<\epsilon$
    $\displaystyle |x^2-9|=|x+3||x-3|<|x-3|^2+6<\epsilon$
    $\displaystyle |x^2-9|=|x+3||x-3|<\delta^2+6<\epsilon$
    $\displaystyle |x^2-9|=|x+3||x-3|<(\delta+6)\delta<\delta^2+6<7\delta<\epsilon$
    The reason for choosing $\displaystyle \delta \leq 1$ is that we know with the choice, $\displaystyle |x^2-9|<\epsilon$ is still true. That does not mean that there is no other value of of $\displaystyle \delta$ besides $\displaystyle 0<\delta\leq 1$. If $\displaystyle 0<\delta\leq 1$ will work, why do we care about larger value of $\displaystyle \delta$? In other words, if the small number is closer to the limit, why bother about other large $\displaystyle \delta$'s that are farther from limit.

    Since we know $\displaystyle 7\delta<\epsilon$ is true, we know $\displaystyle \delta<\frac{\epsilon}{7}$ is also true. So we know it is safe to choose $\displaystyle \delta=\frac{\epsilon}{7}$, since we know with the choice, $\displaystyle |x^2-9|<\epsilon$ will still remain true.

    If we know it will work by limiting $\displaystyle \delta$ to $\displaystyle 0<\delta\leq 1$ to maintain $\displaystyle |X^2-9|<\epsilon$ , and we also know between 0 and 1, the is a number $\displaystyle \frac{\epsilon}{7}$ that also can maintain $\displaystyle |x^2-9|<\epsilon$ being true. Given the choices, of course we would choose $\displaystyle \frac{\epsilon}{7}$ over 1, since the smallest $\displaystyle \delta$ we could get is the one that can give us the value closest to the limit.
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  3. #3
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    Thank you. But how you are sure that

    $\displaystyle
    (6 + \delta)\delta \leq \epsilon
    $

    is true?
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  4. #4
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    Quote Originally Posted by p0oint View Post
    Thank you. But how you are sure that

    $\displaystyle
    (6 + \delta)\delta \leq \epsilon
    $

    is true?
    Yes, it's true.
    $\displaystyle
    (6 + \delta)\delta$ can be as large as $\displaystyle \epsilon$, and it can never reach beyond the value of $\displaystyle \epsilon$. In other words, $\displaystyle (6+\delta)\delta$ can only be made smaller than $\displaystyle \epsilon$ if you choose to make $\displaystyle \delta$ smaller. Because $\displaystyle 7\delta$ is the nearest integer multiple of $\displaystyle \delta$, it's convinient to use 7 for dividing $\displaystyle \epsilon$. Suppose that $\displaystyle \frac{\epsilon}{(6+\delta)\delta}=1$, we can be 100% certain that when $\displaystyle \delta =\frac{\epsilon}{7}$, $\displaystyle (6+\delta)\delta <\epsilon$.

    Example: For when $\displaystyle (6+\delta)\delta = \epsilon$, we can make table as follows:

    $\displaystyle \delta$-------> $\displaystyle x=\delta+3$--------> $\displaystyle (6+\delta)\delta$--------> $\displaystyle \epsilon$

    0.05 ---->3.05------------->0.3025---------->0.3025
    0.50 ---->3.70------------->3.2500---------->3.2500
    010 ----->4.00------------->7.0000---------->7.0000

    The following are the results when $\displaystyle 7 \delta = \epsilon $:

    $\displaystyle \delta$-------> $\displaystyle x=\delta+3$--------> $\displaystyle (6+\delta)\delta$--------> $\displaystyle \epsilon$
    0.05 ---->3.04------------->0.2590---------->0.3021
    0.50 ---->3.42------------->2.7551---------->3.2142
    010 ----->4.00------------->5.8775---------->6.8571

    Now, there should be no doubt the $\displaystyle (6+\delta)\delta \leq \epsilon$.
    Last edited by novice; Nov 5th 2009 at 08:00 AM.
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  5. #5
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    Ok I understand now. Just one thing:

    $\displaystyle |x-3| < \delta $

    and

    $\displaystyle \delta \leq 1 $

    which makes

    $\displaystyle |x-3| \leq 1 $

    $\displaystyle 2 \leq x \leq 4 $

    $\displaystyle 5 \leq x + 3 \leq 7 $

    And we can say $\displaystyle |x+3| \leq 7 $ and it follows $\displaystyle |x+3||x-3| \leq 7|x-3|$

    Because of |x+3|<7 then |x-3||x+3|<7|x-3| and because of |x-3| < delta then 7|x-3|<7delta so that |x-3||x+3|<7|x-3|<7delta

    Now we can set epsilon = 7 delta.

    But one thing why in my book 7 delta <= epsilon ?

    Is it because we can choose smaller values for delta then epsilon/7 ? for ex. delta= epsilon/14 so that epsilon/2 <= epsilon ?
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  6. #6
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    Quote Originally Posted by p0oint View Post
    Ok I understand now. Just one thing:

    $\displaystyle |x-3| < \delta $

    and

    $\displaystyle \delta \leq 1 $

    which makes

    $\displaystyle |x-3| \leq 1 $

    $\displaystyle 2 \leq x \leq 4 $

    $\displaystyle 5 \leq x + 3 \leq 7 $

    And we can say $\displaystyle |x+3| \leq 7 $ and it follows $\displaystyle |x+3||x-3| \leq 7|x-3|$

    Because of |x+3|<7 then |x-3||x+3|<7|x-3| and because of |x-3| < delta then 7|x-3|<7delta so that |x-3||x+3|<7|x-3|<7delta

    Now we can set epsilon = 7 delta.

    But one thing why in my book 7 delta <= epsilon ?

    Is it because we can choose smaller values for delta then epsilon/7 ? for ex. delta= epsilon/14 so that epsilon/2 <= epsilon ?
    For your last question: Yes.
    For the question above it, Yes and No.

    If you chose $\displaystyle \delta$ arbitrarily, $\displaystyle 7\delta \leq \epsilon$ is not necessarily true, but that's not the hypothesis.

    The hypothesis says, for all $\displaystyle \epsilon > 0$, there exist a $\displaystyle \delta > 0$ such that if the distance between $\displaystyle x$ and 3 is less than $\displaystyle \delta$, then $\displaystyle |x^2-9|<\epsilon$. In other words, you must find $\displaystyle \delta$ that is expressed in terms of $\displaystyle \epsilon$---not otherwise.

    Because of the hypothesis, yes, $\displaystyle 7\delta < \epsilon$ is true. In other words, because of the hypothesis, we can choose $\displaystyle \delta$ as small as we like depending on how precise we want, but we cannot choose $\displaystyle \delta$ such that $\displaystyle (6+\delta)\delta > \epsilon$.
    Last edited by novice; Nov 5th 2009 at 12:57 PM.
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  7. #7
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    Quote Originally Posted by p0oint View Post
    Hello!

    I know how to solve this task but I have certain things that I do not understand.

    Solution.

    $\displaystyle |x^2-9|< \epsilon $ if $\displaystyle 0 < |x-3| < \delta$

    Because |x-3| occurs on the right side of this "if statement", it will be helpful to factor the left side to introduce a factor of |x-3|.

    $\displaystyle |x+3||x-3|< \epsilon$ if $\displaystyle 0<|x-3|<\delta$

    Using the triangle inequality:

    $\displaystyle |x+3|=|(x-3)+6| \leq |x-3| + 6$

    By multiplying the statement above with |x-3| (which is positive) we got:

    $\displaystyle |x+3||x-3| \leq (|x-3| +6)|x-3| $

    Now clearly $\displaystyle 6+|x-3|< 6 + \delta$ and $\displaystyle |x-3|<\delta$

    By multiplying both inequalities (I am not sure if this step is valid, please confirm me) I got:

    $\displaystyle (6+|x-3|)|x-3|< (6 + \delta)\delta$

    We can conclude that:

    $\displaystyle |x+3||x-3| < (6 + \delta)\delta$ if $\displaystyle 0 < |x-3| < \delta$

    Now isn't $\displaystyle \epsilon = (6 + \delta)\delta$ ??

    Why in my book they choose $\displaystyle (6 + \delta)\delta \leq \epsilon$ ?

    And why they choose $\displaystyle \delta \geq 1 $ and with that restriction
    $\displaystyle (6 + \delta)\delta \leq 7\delta$ and $\displaystyle 7\delta \leq \epsilon$.

    Finally I don't understand why $\displaystyle \delta = min(e/7,1)$.

    Could somebody possibly explain?

    Thanks a lot.
    The following method was tought by Professor George Thomas of Massachusetts Institute of Technology:

    Given: $\displaystyle |x^2-9|< \epsilon $ if $\displaystyle 0 < |x-3| < \delta$

    Solution:

    $\displaystyle |x^2-9|< \epsilon $
    Expanding the LHS, $\displaystyle |(x+3)(x-3)|< \epsilon $

    Let $\displaystyle 0<\delta < 1$, then $\displaystyle (x+3)$ will be in the neighborhood of 7, and

    $\displaystyle |x-3|<1$----------------------------------eq1

    Say $\displaystyle (x+3)=6.999$, then $\displaystyle |6.999(x-3)|<\epsilon$,
    so that

    $\displaystyle |x-3|<\frac{\epsilon}{7}$--------------------------eq2

    Between eq1 and eq2,

    $\displaystyle \delta = min \{\frac{\epsilon}{7}, 1\}$.
    Last edited by novice; Nov 7th 2009 at 12:33 PM. Reason: fix mistake
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  8. #8
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    Thanks.

    I invented one better method which does not involve arbitary numbers, but I wanted to learn other methods.

    I think you made a little mistake.

    It should be |x-3|<eps./7
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  9. #9
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    Quote Originally Posted by p0oint View Post
    Thanks.

    I invented one better method which does not involve arbitary numbers, but I wanted to learn other methods.

    I think you made a little mistake.

    It should be |x-3|<eps./7
    May I see your invention? Via Private Message if you prefer.
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