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 May 13th, 2019, 06:01 AM #1 Member   Joined: Aug 2016 From: Romania Posts: 32 Thanks: 1 Proof for this power limit without differentiation https://ibb.co/WVC8BgJ I am stuck at the last step and I don't know how to go on.Can someone help me?What do I do from the last step? May 13th, 2019, 01:59 PM #2 Global Moderator   Joined: May 2007 Posts: 6,855 Thanks: 744 In general $\sqrt{1-w}\approx 1-\frac{w}{2}$ for small $w$. Thanks from topsquark May 13th, 2019, 08:40 PM   #3
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 Originally Posted by mathman In general $\sqrt{1-w}\approx 1-\frac{w}{2}$ for small $w$.
How does this avoid differentiation? May 14th, 2019, 08:33 AM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,700 Thanks: 2682 Math Focus: Mainly analysis and algebra \begin{align*} \lim_{x \to \infty} \cos^{x^2} \frac1x &= \lim_{x \to \infty} \left(1 - 2\sin^2\frac1{2x} \right)^{x^2} & (\cos 2A &= 1-2\sin^2 A) \\ &= \lim_{x \to \infty} \left(1 - 2\sin^2\frac1{2x} \right)^{\left(\frac{1}{2\sin^2\frac1{2x}}\right) \left(2{x^2}{\sin^2\frac1{2x}}\right)} \\ &= \lim_{x \to \infty} \left(1 - 2\sin^2\frac1{2x} \right)^{ \frac12 \left( \frac{1}{2\sin^2\frac1{2x}} \right) \left( \frac{\sin\frac1{2x}}{\frac1{2x}} \right)^2} \\ &= \lim_{x \to \infty} \left(1 - 2\sin^2\frac1{2x} \right)^{ \left( \frac{1}{2\sin^2\frac1{2x}} \right) \frac12 \left( \frac{\sin\frac1{2x}}{\frac1{2x}} \right)^2} \\ &= \left( \frac1e \right)^{ \frac12 \cdot 1^2 } \\ &= e^{-\frac12} \end{align*} Thanks from topsquark Last edited by v8archie; May 14th, 2019 at 09:08 AM. May 14th, 2019, 02:22 PM   #5
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Quote:
 Originally Posted by v8archie \begin{align*} \lim_{x \to \infty} \cos^{x^2} \frac1x &= \lim_{x \to \infty} \left(1 - 2\sin^2\frac1{2x} \right)^{x^2} & (\cos 2A &= 1-2\sin^2 A) \\
Where does 1-2sin^2(1/x) come from when cos(1/x) is:,,sqrt(1-sin^2(1/x)''? That part seems confusing. Can you explain how cos(1/x)=cos(2*1/x)?

Last edited by alex77; May 14th, 2019 at 02:24 PM. May 14th, 2019, 02:50 PM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,700 Thanks: 2682 Math Focus: Mainly analysis and algebra The identity used is given in the first line. Tags differentiation, limit, power, proof Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post nekdolan Calculus 7 November 18th, 2014 11:53 PM tuetue Calculus 14 February 10th, 2014 11:38 AM Aikion Calculus 3 March 12th, 2012 10:53 AM jstarks4444 Algebra 1 March 4th, 2011 03:46 AM jstarks4444 Number Theory 0 December 31st, 1969 04:00 PM

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