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Question Number 121302 by liberty last updated on 06/Nov/20

$$\left(1\right)\underset{x\to 0}{\lim }\frac{(1-{\mathrm{e}}^{2\mathrm{x}})\sin \left(3\mathrm{x}\right)}{\mid 4\mathrm{x}\mid }?$$$$\left(2\right)\underset{x\to 0}{\lim }\lfloor \frac{(1-{\mathrm{e}}^{2\mathrm{x}})\sin 3\mathrm{x}}{\mid 4\mathrm{x}\mid }\rfloor ?$$

Answered by bemath last updated on 06/Nov/20

$$\left(1\right)\underset{x\to 0}{\lim }\frac{(1-{\mathrm{e}}^{2\mathrm{x}})\sin 3\mathrm{x}}{\mid 4\mathrm{x}\mid }$$$$\underset{x\to 0^{-}}{\lim }\frac{(1-{\mathrm{e}}^{2\mathrm{x}})\sin 3\mathrm{x}}{-4\mathrm{x}}=$$$$\underset{x\to 0^{-}}{\lim }\frac{3\cos 3\mathrm{x}(1-{\mathrm{e}}^{2\mathrm{x}})-{2\mathrm{e}}^{2\mathrm{x}}\sin 3\mathrm{x}}{-4}=0$$$$\underset{x\to 0^{+}}{\lim }\frac{(1-{\mathrm{e}}^{2\mathrm{x}})\sin 3\mathrm{x}}{4\mathrm{x}}=$$$$\underset{x\to 0^{+}}{\lim }\frac{3\cos 3\mathrm{x}(1-{\mathrm{e}}^{2\mathrm{x}})-{2\mathrm{e}}^{2\mathrm{x}}\sin 3\mathrm{x}}{4}=0$$

Answered by Bird last updated on 06/Nov/20

$$letf\left(x\right)=\frac{(1-e^{2x})sin\left(3x\right)}{4\mid x\mid }$$$$wehave{e}^{2x}\sim 1+2xandsin\left(3x\right)\sim 3x$$$$\Rightarrow f\left(x\right)\sim \frac{-2x\left(3x\right)}{4\mid x\mid }=\frac{-6x^2}{4\mid x\mid }$$$$=-\frac{3}{2}\mid x\mid \Rightarrow {lim}_{x\to 0}f\left(x\right)=0$$

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