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  • The value of <img alt="\lim _{x \rightarrow 0}\left[\frac{x^2}{\sin x \cdot \tan x}\right]" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%200%7D%5Cleft%5B%5Cfr

The value of \lim _{x \rightarrow 0}\left[\frac{x^2}{\sin x \cdot \tan x}\right], where [.] denotes greatest integer function, is

Option: 1

0


Option: 2

1


Option: 3

limit does not exist


Option: 4

2


Answers (1)

best_answer

\begin{aligned} & \mathrm{f}(\mathrm{x})=\mathrm{x}^2-\sin \mathrm{x} \tan \mathrm{x} \\ & \mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{x}-\sin \mathrm{x}\left(\sec ^2 \mathrm{x}+1\right) \\ & \mathrm{f}^{\prime}(\mathrm{x})=2-(\sin \mathrm{x}+\cos \mathrm{x})-2 \sec \mathrm{x} \tan ^2 \mathrm{x} \\ & \sec \mathrm{x}+\cos \mathrm{x}>2 \\ & \therefore \mathrm{f}^{\prime \prime}(\mathrm{x})<0 \Rightarrow \mathrm{f}^{\prime}(\mathrm{x}) \text { is decreasing } \\ & \mathrm{f}^{\prime}(\mathrm{x})<\mathrm{f}(0) \text {, as } \mathrm{x}>0 \\ & \mathrm{f}^{\prime}(\mathrm{x})<0 \Rightarrow \mathrm{f}(\mathrm{x}) \text { is decreasing if } \mathrm{x}>0 \\ & \therefore \mathrm{f}(\mathrm{x})<\mathrm{f}(0) \\ & 0< \frac{\mathrm{x}^2}{\sin \mathrm{x} \tan \mathrm{x}}<1 \end{aligned}

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