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  • If <img alt="\mathrm{\lim _{x \rightarrow 0} \log _{\sec x}(x-\||a|]+\cos x)}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7Bx%20%5Crightarrow%200%7D%20%5Clog%20_%7B

If \mathrm{\lim _{x \rightarrow 0} \log _{\sec x}(x-\||a|]+\cos x)} is -\infty then a lies in the interval (where [.] denotes the greatest integer function )

Option: 1

\mathrm{(-2,-1] \cup[1,2)}


Option: 2

\mathrm{[-2,-1] \cup[1,2]}


Option: 3

\mathrm{[1,2]}


Option: 4

\text { none of these }


Answers (1)

best_answer

\mathrm{ \begin{aligned} & \log _{\sec x}(x-[|a|]+\cos x)=\frac{\ln (x-|| a \mid]+\cos x)}{\ln \sec x} \\\\ & \text { as } x \rightarrow 0, \ln \sec x \rightarrow 0+\text {, so for the limit to be }-\infty, \ln (1-[|a|])<0 \\\\ & \Rightarrow 0<1-[|a|]<1 \end{aligned} }
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Irshad Anwar

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