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Calculus 2: Sequence Limits Question to the power n?

  1. May 3, 2013 #1
    Calculus 2: Sequence Limits Question to the power n??

    1. The problem statement, all variables and given/known data
    Find the limits (if it exists) to decide which sequences, whose nth term is given below.

    2. Relevant equations
    [itex](\frac{3^{n}-4^{n}}{3n^{2}+4^{n}+7})[/itex]

    3. The attempt at a solution
    I've done a few of these but as Soon as the constants were raised to the n. It pretty much stumped me. I haven't done stuff like this yet and I'm sort of self teaching.

    I would appreciate some direction rather than an answer. Thanks!
     
    Last edited: May 3, 2013
  2. jcsd
  3. May 3, 2013 #2

    Zondrina

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    Homework Helper

    Notice the quantity in your numerator is always smaller than zero.

    Notice your denominator is growing without bound.

    So what does the whole thing tend to as n → ∞?
     
  4. May 3, 2013 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    As n grows, exponentials, numbers to the n power, will increase faster than n to a power. And, of course, a larger base will increase faster than a smaller base. That means that [tex]4^n[/tex] will 'dominate' here.

    Divide each term in numerator and denominator by [itex]4^n[/itex] to get
    [tex]\frac{\left(\frac{3}{4}\right)^n- 1}{3\frac{n^2}{4^n}+ 1+ \frac{7}{4^n}}[/tex]

    Now, what does each of those fractions go to?
     
  5. May 3, 2013 #4
    0? I got 0 to start of with but the answer is -1. If that's what you are implying. I have no idea how to mathematically arrive at the answer -1.
     
  6. May 3, 2013 #5
    Negative 1. Got it! I've never done the 'Dominate Term' Approach. Should this be the first approach to consider when doing limits of sequences?

    What's the first things I should consider when I approach questions like this in the future? (Whats the checklist to look out for)

    Thanks man by the way!
     
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