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Question Number 167617 by mathlove last updated on 21/Mar/22

$$\left(1\right)\underset{x\to 0}{\lim }{\left[tan(\frac{\pi }{4}-x)\right]}^{cotx}=?$$$$\left(2\right)\underset{x\to 0}{\lim }[\frac{1}{\sin x}-\frac{1}{x}]=?$$

Answered by cortano1 last updated on 21/Mar/22

$$\left(2\right)\underset{x\to 0}{\lim }\left(\frac{\mathrm{x}-\sin \mathrm{x}}{\mathrm{xsin}\mathrm{x}}\right)=\underset{x\to 0}{\lim }\left(\frac{\mathrm{x}-\sin \mathrm{x}}{{\mathrm{x}}^2\left(\frac{\sin \mathrm{x}}{\mathrm{x}}\right)}\right)$$$$=\underset{x\to 0}{\lim }\left(\frac{\mathrm{x}-\sin \mathrm{x}}{{\mathrm{x}}^2}\right)=\underset{x\to 0}{\lim }\left(\frac{1-\cos \mathrm{x}}{2\mathrm{x}}\right)$$$$=0$$

Answered by LEKOUMA last updated on 21/Mar/22

$$\left(2\right)\underset{x\to 0}{\lim }(\frac{1}{\sin x}-\frac{1}{x})$$$$\underset{x\to 0}{\lim }(\frac{x}{x\sin x}-\frac{\sin x}{x\sin x})=\underset{x\to 0}{\lim }\left(\frac{x-\sin x}{x\sin x}\right)$$$$Hospital$$$$\underset{x\to 0}{\lim }\left(\frac{{(x-\sin x)}^{\prime }}{{\left(x\sin x\right)}^{\prime }}\right)=\underset{x\to 0}{\lim }\left(\frac{{\left(x\right)}^{\prime }-{\left(\sin x\right)}^{\prime }}{{\left(x\sin x\right)}^{\prime }}\right)$$$$\underset{x\to 0}{\lim }\frac{1-\cos x}{\sin x+x\cos x}=\underset{x\to 0}{\lim }\frac{\frac{1-\cos x}{x}}{\frac{\sin x}{x}+\cos x}=\frac{0}{1+1}=\frac{0}{2}=0$$$$\underset{x\to 0}{\lim }(\frac{1}{\sin x}-\frac{1}{x})=0$$

Answered by qaz last updated on 21/Mar/22

$$\underset{\mathrm{x}\to 0}{\lim }{\left[\tan (\frac{\pi }{4}-\mathrm{x})\right]}^{\cot \mathrm{x}}=\underset{\mathrm{x}\to 0}{\lim }{\left(\frac{\cot \mathrm{x}-1}{\cot \mathrm{x}+1}\right)}^{\cot \mathrm{x}}=\underset{\mathrm{x}\to 0}{\lim }{\left[{(1-\frac{2}{\cot \mathrm{x}+1})}^{\frac{\cot \mathrm{x}+1}{2}}\right]}^2{(1-\frac{2}{\cot \mathrm{x}+1})}^{-1}={\mathrm{e}}^{-2}$$

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