Chủ đề này có 1 trả lời

[Enzan]

Tinh: $\lim_{x\rightarrow 0}\frac{\sqrt{1+2x}-cosx-x}{x^2}$

[E. Galois]

Ta có:

\begin{align*}\lim_{x\rightarrow 0}\frac{\sqrt{1+2x}-cosx-x}{x^2} &=\lim_{x\rightarrow 0}\left (\frac{\sqrt{1+2x}-x-1}{x^2}+\frac{1-\cos x}{x^2}  \right ) \\&=\lim_{x\rightarrow 0}\left (\frac{-x^2}{x^2(\sqrt{1+2x}+x+1)}+\frac{2\sin^2\frac{x}{2}}{x^2}  \right ) \\&= \lim_{x\rightarrow 0}\left (\frac{-1}{\sqrt{1+2x}+x+1}+\frac{1}{2}.\frac{\sin^2\frac{x}{2}}{\left (\frac{x}{2}  \right )^2}  \right ) \\&=-\frac{1}{2}+\frac{1}{2}\\&=0 \end{align*}