Power Series and Limits

Key Questions

  • How do I find lim_(x->oo)(3sin(x))/e^x using power series?

    To be honest, I would not use power series on this one since this is a perfect problem to demonstrate the application of Squeeze Theorem. Here is how:

    We know

    -1 le sinx le 1

    Rightarrow -3 le 3sinx le 3

    Rightarrow -3/e^x le {3sinx}/e^x le 3/e^x.

    Since

    lim_{x to infty}(-3/e^x)=-3/infty=0

    and

    lim_{x to infty}3/e^x=3/infty=0,

    we conclude that

    lim_{x to infty}{3sinx}/e^x=0

    by Squeeze Theorem.

    I hope that this was helpful.

    Wataru · 1 · Oct 19 2014
  • How do I use a power series to calculate a limit?

    Here is a simple application of a power series in evaluating a limit.

    lim_{x to 0}{sinx]/x

    by replacing sinx by its Maclaurin series.

    =lim_{x to 0}{x-x^3/{3!}+x^5/{5!}-x^7/{7!}+cdots}/{x}

    by distributing the division to each term,

    =lim_{x to 0}(1-x^2/{3!}+x^4/{5!}-x^6/{7!}+cdots)

    by sending x to zero,

    =1-0+0-0+cdots

    since all but the first term are zero,

    =1

    I hope that this was helpful.

    Wataru · · Oct 19 2014

Questions

Power Series