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Question Number 82924 by jagoll last updated on 26/Feb/20

$$\underset{x\to 0}{\lim }\frac{\sqrt{\tan \mathrm{x}}-\sqrt{\sin \mathrm{x}}}{{\mathrm{x}}^2\sqrt{\mathrm{x}}}$$

Commented by jagoll last updated on 26/Feb/20

$$\mathrm{the}\mathrm{ans}:\frac{1}{4}$$

Commented by john santu last updated on 26/Feb/20

$$\underset{x\to 0}{\lim }\frac{\tan \mathrm{x}-\sin \mathrm{x}}{{\mathrm{x}}^2\sqrt{\mathrm{x}}(\sqrt{\tan \mathrm{x}}+\sqrt{\sin \mathrm{x}})}=$$$$\underset{x\to 0}{\lim }\frac{\tan \mathrm{x}(1-\cos {}^2\mathrm{x})}{{\mathrm{x}}^2\sqrt{\mathrm{x}}(\sqrt{\mathrm{x}}+\sqrt{\mathrm{x}})}=$$$$\mathrm{note}\underset{x\to 0}{\lim }\tan \mathrm{x}=\underset{x\to 0}{\lim }\sin \mathrm{x}\approx \underset{x\to 0}{\lim }\mathrm{x}$$$$\underset{x\to 0}{\lim }\frac{\mathrm{x}.\left(\frac{1}{2}{\mathrm{x}}^2\right)}{{2\mathrm{x}}^3}=\left(\frac{\left(\frac{1}{2}\right)}{2}\right)=\frac{1}{4}$$$$$$

Answered by TANMAY PANACEA last updated on 26/Feb/20

$$\underset{x\to 0}{\lim }\frac{\sqrt{sinx}\times (\frac{1}{\sqrt{cosx}}-1)}{x^2\times \sqrt{x}}$$$$\underset{x\to 0}{\lim }{\left(\frac{sinx}{x}\right)}^{\frac{1}{2}}\times \frac{1-cosx}{x^2}\times \frac{1}{1+\sqrt{cosx}}$$$$\underset{x\to 0}{\lim }{\left(\frac{sinx}{x}\right)}^{\frac{1}{2}}\times \frac{2sin\frac{x}{2}\times sin\frac{x}{2}}{\frac{x}{2}\times \frac{x}{2}\times 4}\times \frac{1}{1+\sqrt{cosx}}$$$$\underset{x\to 0}{\lim }{\left(\frac{x-\frac{x^3}{3!}+\frac{x^5}{5!}+..}{x}\right)}^{\frac{1}{2}}\times \frac{1}{2}\times {\left(\frac{sin\frac{x}{2}}{\frac{x}{2}}\right)}^2\times \frac{1}{1+\sqrt{cosx}}$$$$=1\times \frac{1}{2}\times 1\times \frac{1}{2}=\frac{1}{4}$$

Commented by jagoll last updated on 26/Feb/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

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