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Question Number 82988 by jagoll last updated on 26/Feb/20

$$\underset{x\to 0}{\lim }\frac{1-\sqrt{1+{\mathrm{x}}^2}\cos \mathrm{x}}{{\mathrm{x}}^4}$$

Commented by jagoll last updated on 26/Feb/20

$$\mathrm{my}\mathrm{answer}\mathrm{is}\frac{1}{3}.\mathrm{that}\mathrm{is}\mathrm{correct}?$$

Commented by mr W last updated on 26/Feb/20

$$=\frac{1-(1+\frac{x^2}{2}-\frac{x^4}{8}+o\left(x^4\right))(1-\frac{x^2}{2}+\frac{x^4}{24}+o\left(x^4\right))}{x^4}$$$$=\frac{1-(1-\frac{x^4}{3}+o\left(x^4\right))}{x^4}$$$$=\frac{\frac{x^4}{3}+o\left(x^4\right)}{x^4}$$$$=\frac{1}{3}$$

Commented by jagoll last updated on 27/Feb/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

Answered by jagoll last updated on 26/Feb/20

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