
PreCalculus PreCalculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
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 $\sqrt{1w}\approx 1\frac{w}{2}$ for small $w$.

May 13th, 2019, 08:40 PM  #3 
Senior Member Joined: Sep 2016 From: USA Posts: 684 Thanks: 459 Math Focus: Dynamical systems, analytic function theory, numerics  
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 &= 12\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*} Last edited by v8archie; May 14th, 2019 at 09:08 AM. 
May 14th, 2019, 02:22 PM  #5 
Member Joined: Aug 2016 From: Romania Posts: 32 Thanks: 1  Where does 12sin^2(1/x) come from when cos(1/x) is:,,sqrt(1sin^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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
What is the limit of this power law function?  nekdolan  Calculus  7  November 18th, 2014 11:53 PM 
Problem with squareroot and power differentiation  tuetue  Calculus  14  February 10th, 2014 11:38 AM 
differentiation of e with a fractional power  Aikion  Calculus  3  March 12th, 2012 10:53 AM 
Power Proof  jstarks4444  Algebra  1  March 4th, 2011 03:46 AM 
Power Proof  jstarks4444  Number Theory  0  December 31st, 1969 04:00 PM 