Saturday, 14 May 2016

calculus - Evaluating limxto0fracsin(4x)sin(3x) without L'Hospital




I need to evaluate
limx0sin(4x)sin(3x)
I solved this using L'Hospital's Theorem and I got 4/3
However, is there a way to do this without applying this theoerm?


Answer



As \lim_{x \to 0}\frac{\sin(x)}{x}=1
\lim_{x \to 0}{\frac{\sin(4x)}{\sin(3x)}} can be written as
\frac{4}{3}\lim_{x \to 0}\frac{\sin(4x)}{4x}\frac{3x}{\sin(3x)}
=\frac{4}{3}



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