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Math Help - evaluating limits of trig functions

  1. #1
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    evaluating limits of trig functions

    Evalue each of the following limits.

    Ok I am having trouble with evaluating trig limits with the tan function. I was wondering if someone could show me step by step their answer for the following three questions:

    a) lim (tanx)/(4x)
    x-> 0

    b) lim (x^3)/(tan^3(2x))
    x->0

    c) lim (2tanx)/(xsecx)
    x->0

    please explain it thoroughly as you can. Appreciate it very much. Thanks.
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  2. #2
    Super Member Random Variable's Avatar
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    a)  \lim_{x \to 0} \ \frac{\tan x}{4x} = \frac{1}{4} \lim_{x \to 0} \ \frac{\sin x}{x \cos x} = \frac{1}{4} \Big( \lim_{x \to 0} \ \frac{1}{\cos x} \cdot \lim_{x \to 0} \ \frac{\sin x}{x}\Big) = \frac{1}{4} \Big(1 \cdot 1\Big) = \frac{1}{4}
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  3. #3
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    Quote Originally Posted by Random Variable View Post
    a)  \lim_{x \to 0} \ \frac{\tan x}{4x} = \frac{1}{4} \lim_{x \to 0} \ \frac{\sin x}{x \cos x} = \frac{1}{4} \Big( \lim_{x \to 0} \ \frac{1}{\cos x} \cdot \lim_{x \to 0} \ \frac{\sin x}{x}\Big) = \frac{1}{4} \Big(1 \cdot 1\Big) = \frac{1}{4}
    question.. when you break the eqtn into two, shown by the two limits at
    1/cosx and sinx/x.. this step: isn't (1 x 1) wrong because sin0/0 is still undefined?
    sorry, i'm not understanding that part
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  4. #4
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    actually I have it now, question 1 anyways.

    For question two, this is what I have so far:

    lim x^3/ [tan^(3) 2x]
    x->0
    = x^3 [ cos^(3) x/ sin^(3) x]
    not sure what move to make next
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