Tuesday, 17 June 2014

calculus - Find limxto0fraccos2x1cosx1 without L'Hopital's rule.



limx0cos2x1cosx1

I have found the above limit using L'Hopital's rule but since this rule is not given in the book so I'm supposed to do it without using this rule.



I know \lim_{x \to 0}\frac{1-\cos x}{x}=0



I tried to get something of the form of the above limit but I failed to do so.



Kindly help me solve this problem without using L'Hopital's rule.


Answer



Recall \cos(2x)=2\cos^2(x)-1 so we may rewrite as




\lim\limits_{x\to 0} 2\frac{\cos^2(x)-1}{\cos(x)-1}=\lim\limits_{x\to 0} 2\frac{(\cos(x)-1)(\cos(x)+1)}{\cos(x)-1}=2(\cos(0)+1)=4


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