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More limits! \begin{alignat*}{3} &a)   \lim_{x \to 0}{\frac{\tan {3x} }{ x}} &             &b)    \lim_{x \to 0} {\frac{\sec {2x} \tan {2x} }{ x}} &             &c)    \lim_{t \to 0^{+}} \sin (\sqrt x \cdot x)\\ &d)    \lim_{x \to 0}{\frac{1-\cos {2x} }{ x}} &             &e)    \lim_{x \to 0}{\frac{1-\cos {2x} }{ x^2}} \end{alignat*}

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Tags: trig-limits, limits, limits-nonexistence, trig-identities, trig, root, composite, rote-problem, tricks, computation, a--a, t2, c1

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This collects a few trig limit tricks. (a) (b) and (d) just involve rewriting the expressions to use “sin x / x” and “1 - cos x / x”. (e) is another multiply by the conjugate (1 + cos 2x) problem and (c) is straightforward. This problem might be used to encourage automaticity.

– Eric Hsu 1998-08-09