Thursday, 17 January 2013

calculus - find this limit without l'hopital's rule



I can't figure out how to get the limit in this problem. I know that 1cosxx=0 but I'm not allowed to use L'Hopital's Rule. I also already know that the answer is 2536 but I don't know the steps in between. I've already tried multiplying by the conjugates of both the numerator and the denominator but neither are getting me anywhere close. Here is the question:



lim



Answer



Outline: Our expression is equal to
-\frac{1+\cos 6x}{1+\cos 5x}\cdot\frac{1-\cos^25x}{1-\cos^2 6x},\tag{1}
which is
-\frac{1+\cos 6x}{1+\cos 5x}\cdot\frac{\sin^2 5x}{\sin^2 6x}.\tag{2}
To find
\lim_{x\to 0} \frac{\sin 5x}{\sin 6x}, rewrite as
\frac{5}{6}\lim_{x\to 0} \frac{\frac{\sin 5x}{5x}}{\frac{\sin 6x}{6x}}.


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