My Math Forum Proof for this power limit without differentiation

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 May 13th, 2019, 05:01 AM #1 Newbie   Joined: Aug 2016 From: Romania Posts: 23 Thanks: 0 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, 12:59 PM #2 Global Moderator   Joined: May 2007 Posts: 6,755 Thanks: 695 In general $\sqrt{1-w}\approx 1-\frac{w}{2}$ for small $w$. Thanks from topsquark
May 13th, 2019, 07:40 PM   #3
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Quote:
 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, 07:33 AM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 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 08:08 AM.
May 14th, 2019, 01:22 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 01:24 PM.

 May 14th, 2019, 01:50 PM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra The identity used is given in the first line.

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