Saturday, 16 December 2017

calculus - Solving limlimitsxto0fracxsin(x)x2 without L'Hospital's Rule.



How to solve lim Without L'Hospital's Rule?
you can use trigonometric identities and inequalities, but you can't use series or more advanced stuff.


Answer



The given expression is odd; therefore it is enough to consider x>0. We then have
0<{x-\sin x\over x^2}<{\tan x -\sin x\over x^2}=\tan x\ {1-\cos x\over x^2}={\tan x\over2}\ \Bigl({\sin(x/2)\over x/2}\Bigr)^2\ ,
and right side obviously converges to 0 when x\to0+.


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