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Limits Question

  1. Sep 20, 2009 #1
    1. The problem statement, all variables and given/known data
    [tex]Let \[f(x)=\begin{cases}{}
    \frac{6a(x^2+1)}{2x^2+1}+\frac{b\log \big((x+1)^4\big)}{x} &\mbox{if } x<0,\\
    -2a-4 &\mbox{if } x=0, \\
    \frac{a\sin x}{x}+b &\mbox{if } x>0.
    \end{cases}\]
    Find \(a\) and \(b\) such that \(f\) is continuous at \(x=0\).[/tex]

    2. Relevant equations
    Left and Right Hand limits are equal, as well as limit laws (If you're helping with the question I assume you know what they are)

    3. The attempt at a solution
    If f is continuous at x=0, the limit as x approaches 0 from the left should be equal to the limit as x approaches 0 from the right. With that, I took the limit of the part of f where x<0 and set it equal to the limit of the part of f where x>0.

    Using L'hopital's rule I computed: [tex]\lim_{x \to 0-}\frac{4b\log{x+1}}{x}[/tex] and the result is 4b*log(e). If I set this equal to the right hand limit, I can calculate that [tex]b=\frac{-5a}{4\log{e}-1}[/tex]. Now I am stuck and am not sure what to do. Thanks for the help.
     
  2. jcsd
  3. Sep 20, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi JBD2! :smile:

    (I expect log here means log base e :wink:)

    Now solve for b/a, use that to get rid of b, and make one of the limits equal to -2a - 4. :smile:
     
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