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Is this right? (limits w/ trig)

  1. Oct 26, 2006 #1
    Determine whether f'(0) exists.
    f(x) = { x2sin (1/x) if x does not equal 0, 0 if x = 0}

    [tex]f'(0) = \lim_{x\rightarrow 0} \frac{x^2 \sin \frac{1}{x}}{x - 0}[/tex]
    [tex]f'(0) = \lim_{x\rightarrow 0} \frac{\sin \frac{1}{x}}{\frac{1}{x}}[/tex]
    Let u = 1/x. As x approaches 0, u approaches infinity. So I can't use lim x->0 (sin x)/x = 1. But now I have:
    [tex]f'(0) = \lim_{u\rightarrow \infty} \frac{sin u}{u}[/tex]
    since sin u oscillates between -1 and 1, and u goes to infinity, f'(0) = 0.
    Last edited: Oct 26, 2006
  2. jcsd
  3. Oct 26, 2006 #2
    sure that is fine, you could also note that -x^2<=x^2*sin(1/x)<=x^2 and use that.
    Last edited: Oct 26, 2006
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