Sunday, 16 November 2014

calculus - Evaluate limxrightarrow0fracsinxx+tanx without L'Hopital



I need help finding the the following limit:




lim



I tried to simplify to:



\lim_{x\rightarrow 0} \frac{\sin x \cos x}{x\cos x+\sin x}



but I don't know where to go from there. I think, at some point, you have to use the fact that \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1. Any help would be appreciated.



Thanks!


Answer




\frac{\sin x}{x + \tan x} = \frac{1}{\frac{x}{\sin x}+\frac{\tan x}{\sin x}} \to 1/2


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