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Question Number 71360 by 20190927 last updated on 14/Oct/19

$$\underset{x\to 0}{\lim }\frac{1-\sqrt{1+{2\mathrm{x}}^4}\cos \left(\sqrt{2}{\mathrm{x}}^2\right)}{{\mathrm{x}}^5\ln (1-{2\mathrm{x}}^3)}$$

Commented by mathmax by abdo last updated on 14/Oct/19

$$forx\in V\left(0\right)\sqrt{1+2x^4}\sim 1+x^{4}andcos\left(\sqrt{2}{x}^2\right)\sim 1-\frac{2x^4}{2}=1-x^4\Rightarrow$$$$1-\sqrt{1+2x^4}cos\left(\sqrt{2}{x}^2\right)\sim 1-(1+x^4)(1-x^4)=1-(1-x^8)=x^8$$$$ln(1-2x^3)\sim -2x^3\Rightarrow x^{5}ln(1-2x^3)\sim -2x^8\Rightarrow$$$${lim}_{x\to 0}\frac{1-\sqrt{1+2x^4}cos\left(\sqrt{2}{x}^2\right)}{x^{5}ln(1-2x^3)}=-\frac{1}{2}$$

Commented by 20190927 last updated on 17/Oct/19

$$\mathrm{thanks}$$

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