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Question Number 83019 by mathmax by abdo last updated on 27/Feb/20

$$find{lim}_{x\to 0}\frac{1-\sqrt{1+x+x^2}cos\left(2x\right)}{x^3}$$

Commented by mathmax by abdo last updated on 27/Feb/20

$$letf\left(x\right)=\frac{1-\sqrt{1+x+x^2}cos\left(2x\right)}{x^3}wehavecos\left(2x\right)\sim 1-2x^{2}and$$$$\sqrt{1+x+x^2}\sim 1+\frac{1}{2}(x+x^2)\Rightarrow 1-\sqrt{1+x+x^2}cos\left(2x\right)$$$$\sim 1-(1+\frac{1}{2}x+\frac{1}{2}{x}^2)(1-2x^2)$$$$=1-(1-2x^2+\frac{x}{2}-x^3+\frac{x^2}{2}-x^4)=2x^2-\frac{x}{2}+x^3-\frac{x^2}{2}+x^4$$$$=x^4+x^3+\frac{3}{2}{x}^2-\frac{x}{2}\Rightarrow f\left(x\right)\sim x+1+\frac{3}{2x}-\frac{1}{2x^2}\Rightarrow$$$${lim}_{x\to 0}f\left(x\right)=\mathrm{\infty }$$

Answered by john santu last updated on 27/Feb/20

$$\underset{x\to 0}{\lim }\frac{1-\sqrt{1+\mathrm{x}+{\mathrm{x}}^2}(1-2\sin {}^2\mathrm{x})}{{\mathrm{x}}^3}$$$$\underset{x\to 0}{\lim }\frac{1-\sqrt{1+\mathrm{x}+{\mathrm{x}}^2}}{{\mathrm{x}}^3}+\underset{x\to 0}{\lim }\frac{2\sqrt{1+\mathrm{x}+{\mathrm{x}}^2}\sin {}^2\mathrm{x}}{{\mathrm{x}}^3}$$$$\underset{x\to 0}{\lim }\frac{-{\mathrm{x}}^2-\mathrm{x}}{{2\mathrm{x}}^3}+\underset{x\to 0}{\lim }\frac{2}{\mathrm{x}}{\left(\frac{\sin \mathrm{x}}{\mathrm{x}}\right)}^2=\mathrm{\infty }$$

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