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Limit proofs

  1. May 21, 2006 #1
    Hey, I'm stuck on this problem and just wanted some help if possible.

    Prove, using an [tex]\epsilon - \delta[/tex] argument, that [tex]\lim_{\substack{x\rightarrow -2}} \frac {1 - x - 2x^2} {x + 1} = 5[/tex]

    Ok, so I've answered so far with:

    [tex]\lim_{\substack{x\rightarrow -2}} \frac {1 - x - 2x^2} {x + 1} = 5[/tex] if [tex]\forall \epsilon > 0, \exists \delta > 0 : 0 < |x + 2| < \delta \Rightarrow |-2x - 4| < \epsilon[/tex]

    Investigation to find [tex]\delta[/tex]
    [tex]|-2x - 4| < \epsilon[/tex]
    [tex]\Rightarrow -2|x + 2| < \epsilon[/tex]
    [tex]\Rightarrow |x + 2| > -\frac {\epsilon} {2}[/tex]

    Therefore, choose [tex]\delta = -\frac {\epsilon} {2}[/tex]

    My problem is here, my statement was that [tex]|x + 2| < \delta[/tex], but my answer so far has given me [tex]|x + 2| > \delta[/tex]. Also, [tex]\delta[/tex] is [tex]< 0[/tex].

    What should I change?
     
    Last edited: May 21, 2006
  2. jcsd
  3. May 21, 2006 #2

    LeonhardEuler

    User Avatar
    Gold Member

    Here's where you made the mistake:
    Think about it: |-2x - 4| must be greater than or equal to zero, while -2|x+2| must be less than or equal to zero. They can not possibly be equal for all x.
     
  4. May 22, 2006 #3
    I don't really understand what you're getting at...

    edit: Could I do this?
    [tex]\begin{align*}
    |-2x - 4| < \epsilon\\
    \Rightarrow -2|x + 2| < \epsilon\\
    \Rightarrow |x + 2| > -\frac {\epsilon} {2}\\
    \Leftrightarrow |x + 2| < \frac {\epsilon} {2} \ or\ |x + 2| > -\frac {\epsilon} {2}
    \end{align*}[/tex]

    Therefore, choose [tex]\delta = \frac {\epsilon} {2}[/tex]

    Proof:
    Let [tex]\epsilon > 0[/tex] and set [tex]\delta = \frac {\epsilon} {2}[/tex]

    [tex]\begin{align*}
    |x + 2| < \delta\\
    \Rightarrow |x + 2| < \frac {\epsilon} {2}\\
    \Leftrightarrow -\frac {\epsilon} {2} < |x + 2| < \frac {\epsilon} {2}\\
    \Rightarrow |x + 2| > -\frac {\epsilon} {2}\\
    \Rightarrow -2|x + 2| < \epsilon\\
    \Rightarrow |-2x - 4| < \epsilon
    \end{align*}[/tex]
    As required.
     
    Last edited: May 22, 2006
  5. May 22, 2006 #4

    matt grime

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    Science Advisor
    Homework Helper

    The point is that you should not pull out negative numbers from inside an abs value symbol and leave them as negative: |-2x-4| = 2|x+2|.

    It is 'vacuous' to say that -2|x+2|<e in the sense that it is always true since the left hand side is always negative and the right hand side is positive by assumption.

    In effect you have gone through several unnecessary and dubious steps to conclude that |x+2|<e/2 which you could have done straight away by noting that |-2x-4|=2|x+2|. I say dubious because you have concluded that either this holds OR that |x+2|<e/-2, which is always true (for positive e) hence you have not got any restriction on x at all.
     
  6. May 22, 2006 #5
    Ok, I fixed it up to make it look like this:


    Prove, using an [tex]\epsilon - \delta[/tex] argument, that [tex]\lim_{\substack{x\rightarrow -2}} \frac {1 - x - 2x^2} {x + 1} = 5[/tex]


    [tex]\lim_{\substack{x\rightarrow -2}} \frac {1 - x - 2x^2} {x + 1} = 5[/tex] if [tex]\forall \epsilon > 0, \exists \delta > 0 : 0 < |x + 2| < \delta \Rightarrow |-2x - 4| < \epsilon[/tex]

    Investigation to find [tex]\delta[/tex]
    [tex]\begin{align*}
    |-2x - 4| < \epsilon\\
    \Rightarrow |2x + 4| < \epsilon\\
    \Rightarrow 2|x + 2| < \epsilon\\
    \Rightarrow |x + 2| < \frac {\epsilon} {2}
    \end{align*}[/tex]

    Therefore, choose [tex]\delta = \frac {\epsilon} {2}[/tex]

    Proof:
    Let [tex]\epsilon > 0[/tex] and set [tex]\delta = \frac {\epsilon} {2}[/tex]

    [tex]\begin{align*}
    |x + 2| < \delta\\
    \Rightarrow |x + 2| < \frac {\epsilon} {2}\\
    \Rightarrow 2|x + 2| < \epsilon\\
    \Rightarrow |2x + 4| < \epsilon\\
    \Rightarrow |-2x - 4| < \epsilon
    \end{align*}[/tex]
    As required
     
    Last edited: May 22, 2006
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