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Question Number 111259 by bobhans last updated on 03/Sep/20

$$\underset{x\to 0^{+}}{\lim }\frac{\sin 2\mathrm{x}}{\sqrt{3\mathrm{x}}}?$$

Commented by bobhans last updated on 03/Sep/20

$$\underset{x\to 0^{+}}{\lim }\frac{\sin 2\mathrm{x}}{\sqrt{3\mathrm{x}}}.\frac{\sqrt{3\mathrm{x}}}{\sqrt{3\mathrm{x}}}=\underset{x\to 0^{+}}{\lim }\frac{\sqrt{3\mathrm{x}}\sin 2\mathrm{x}}{3\mathrm{x}}$$$$=\underset{x\to 0^{+}}{\lim }\frac{2\sqrt{3\mathrm{x}}}{3}\times \underset{x\to 0^{+}}{\lim }\frac{\sin 2\mathrm{x}}{2\mathrm{x}}=0\times 1=0$$

Answered by abdomsup last updated on 03/Sep/20

$$ch.\sqrt{x}=tgive{lim}_{x\to 0^{+}}\frac{sin\left(2x\right)}{\sqrt{3}\sqrt{x}}$$$$={lim}_{t\to 0^{+}}\frac{sin\left(2t^2\right)}{t\sqrt{3}}$$$$={lim}_{t\to 0^{+}}\frac{2t}{\sqrt{3}}\times \frac{sin\left(2t^2\right)}{2t^2}$$$$=0\times 1=0$$

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