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Question Number 109068 by n0y0n last updated on 20/Aug/20

Answered by Dwaipayan Shikari last updated on 20/Aug/20

$$\underset{x\to 0}{\lim }{\left(cos5x^2\right)}^{\frac{1}{x^4}}={(1-\frac{25x^4}{2})}^{\frac{-2}{x^4.25}.\left(\frac{-25}{2}\right)}=e^{-\frac{25}{2}}$$

Answered by Dwaipayan Shikari last updated on 20/Aug/20

$$\underset{x\to 0}{\lim }{\left(cos5x^2\right)}^{\frac{1}{x^4}}=y$$$$\frac{1}{x^4}log\left(cos5x^2\right)=logy$$$$\frac{1}{x^4}log(1+cos5x^2-1)=logy$$$$\frac{cos5x^2-1}{x^4}.\frac{log(1+cos5x^2-1)}{cos5x^2-1}=logy$$$$\frac{-2{sin}^2\frac{5x^2}{2}}{x^4}=logy$$$$-\frac{25}{2}=logy$$$$y=e^{-\frac{25}{2}}$$

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