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Question Number 149251 by john_santu last updated on 04/Aug/21

$$\underset{x\to 0^{+}}{\lim }\frac{\tan \mathrm{x}\sqrt{\tan \mathrm{x}}-\sin \mathrm{x}\sqrt{\sin \mathrm{x}}}{{\mathrm{x}}^3\sqrt{\mathrm{x}}}=?$$$$$$

Answered by dumitrel last updated on 04/Aug/21

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

Answered by john_santu last updated on 04/Aug/21

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