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Question Number 126195 by bramlexs22 last updated on 18/Dec/20

$$\underset{x\to 1}{\lim }\frac{2\sqrt{x}-\sin (x-1)-2\cos (x-1)}{\arctan (x-1)-\ln x}?$$

Answered by liberty last updated on 18/Dec/20

$$letw=x-1$$$$L=\underset{w\to 0}{\lim }\frac{2\sqrt{w+1}-\sin w-2\cos w}{\arctan w-\ln (w+1)}$$$$L=\underset{w\to 0}{\lim }\frac{2(1+\frac{w}{2}-\frac{w^2}{8}+O\left(w^3\right))-(w+O\left(w^3\right))-2(1-\frac{w^2}{2}+O\left(w^4\right))}{(w+O\left(w^3\right))-(w-\frac{w^2}{2}+O\left(w^3\right))}$$$$L=\underset{w\to 0}{\lim }\frac{\frac{3w^2}{4}+O\left(w^3\right)}{\frac{w^2}{2}+O\left(w^3\right)}=\underset{w\to 0}{\lim }\frac{w^2(\frac{3}{4}+O\left(w\right))}{w^2(\frac{1}{2}+O\left(w\right))}$$$$L=\frac{\frac{3}{4}+0}{\frac{1}{2}+0}=\frac{3}{2}$$

Answered by bramlexs22 last updated on 18/Dec/20

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