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Homework Help: Function with denominator zero

  1. Jul 27, 2009 #1
    1. The problem statement, all variables and given/known data

    This is a limit problem but what I need to figure out is simpler so I thought I'd post it under pre-calc. The question is:

    Find the limit:

    lim as x approaches 0 of (tan 2x)/x

    2. Relevant equations

    3. The attempt at a solution

    Since x is in the denominator I know that I must re-write (tan 2x)/x so that the denominator doesn't equal zero. I also know that 0 is a root of both the numerator and denominator, but I don't know how to re-write such an equation. Any help? Thanks!
  2. jcsd
  3. Jul 27, 2009 #2
    Use L'Hopital's Rule
  4. Jul 27, 2009 #3


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    Science Advisor
    Homework Helper

    If you haven't done l'Hopital yet, do you know lim x->0 tan(x)/x=1 or lim x->0 sin(x)/x=1? Then you could just do the variable substitution u=2x.
  5. Jul 28, 2009 #4
    Why can't we obtain the limit here by squeezing?

    I'll show you a similar example with tan(x) instead of (tan 2x):

    [tex]\lim_{x \to 0} \frac{tanx}{x} = \lim_{x \to 0} (\frac{sin x}{x}.\frac{1}{cos x})[/tex]

    [tex](\lim_{x \to 0} \frac{sin x}{x})(\lim_{x \to 0} \frac{1}{cos x}) = (1)(1) =1[/tex]

    In your question you must do some manipulation for the 2 in tan2x.
  6. Jul 28, 2009 #5


    Staff: Mentor

    You're not actually doing any "squeezing" here--just using the fact that [tex]\lim_{x \to 0} \frac{sin x}{x} = 1[/tex]
    This limit is often proved by the "squeeze-play" theorem, but can be done other ways.
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