Question

lim tgx-senx/sen²x
x->0

Answer 1

            $L=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{\sin^2 x}\\ \\ \\ L=\lim\limits_{x\to 0}\dfrac{\dfrac{\sin x}{\cos x}-\sin x}{\sin^2 x}\\ \\ \\ L=\lim\limits_{x\to 0}\dfrac{1-\cos x}{\sin x\cos x}\\ \\ \\ L=\lim\limits_{x\to 0}\dfrac{(1-\cos x)(1+\cos x)}{\sin x\cdot \cos x\cdot (1+\cos x)}\\ \\ \\ L=\lim\limits_{x\to 0}\dfrac{1-\cos^2 x}{\sin x\cdot \cos x\cdot (1+\cos x)}$

           $L=\lim\limits_{x\to 0}\dfrac{\sin^2 x}{\sin x\cdot \cos x\cdot (1+\cos x)}\\ \\ \\ L=\lim\limits_{x\to 0}\dfrac{\sin x}{\cos x\cdot (1+\cos x)}\\ \\ \\ L=\dfrac{0}{1\cdot 2}\\ \\ \boxed{L=0}$