Saturday, July 6, 2019

calculus - Find limlimitsxto0fraclogleft(cosxright)x2 without L'Hopital



lim



I've been triyng to:




  1. show $\displaystyle -\frac{\pi}{2}


  2. find a function so that \displaystyle f(x)<\frac{\log\left(\cos x\right)}{x^2} and \displaystyle \lim\limits_{x\to0}f(x) = -\frac{1}{2}





And then apply the squeeze principle, but haven't managed any of these.


Answer



HINT:



\dfrac{\log(\cos x)}{x^2}=\dfrac{\log(\cos^2x)}{2x^2}=-\dfrac12\cdot\dfrac{\log(1-\sin^2x)}{-\sin^2x}\cdot\left(\dfrac{\sin x}x\right)^2


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