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Question Number 112311 by bemath last updated on 07/Sep/20

$$\underset{x\to 0}{\lim }\sqrt{\frac{3\sin {}^2\mathrm{x}+\cos 2\mathrm{x}-1}{\mathrm{x}.\tan 2\mathrm{x}}}?$$

Commented by Dwaipayan Shikari last updated on 07/Sep/20

$$\underset{x\to 0}{\lim }\sqrt{\frac{3{sin}^{2}x+1-2{sin}^{2}x-1}{x\left(2x\right)}}=\sqrt{\frac{{\left(x\right)}^2}{2x^2}}=\frac{1}{\sqrt{2}}(tanx\to x,sinx\to x)$$

Commented by malwan last updated on 07/Sep/20

$$Whatif\underset{x\to 0^{-}}{lim}?$$

Answered by mathmax by abdo last updated on 07/Sep/20

$$\mathrm{let}\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\frac{{3\sin }^2\mathrm{x}+\cos \left(2\mathrm{x}\right)-1}{\mathrm{xtan}\left(2\mathrm{x}\right)}}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)\sim \sqrt{\frac{{3\mathrm{x}}^2+1-\frac{{4\mathrm{x}}^2}{2}-1}{{2\mathrm{x}}^2}}$$$$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)\sim \sqrt{\frac{{\mathrm{x}}^2}{{2\mathrm{x}}^2}}=\frac{1}{\sqrt{2}}\Rightarrow {\lim }_{\mathrm{x}\to 0}\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\sqrt{2}}$$

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