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  • <img alt="\mathrm{Among\, \, \lim _{x \rightarrow 0} \sec ^{-1}\left(\frac{x}{\sin x}\right) }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BAmong%5C%2C%20%5C%2C%20%5Clim%20_%7B

\mathrm{Among\, \, \lim _{x \rightarrow 0} \sec ^{-1}\left(\frac{x}{\sin x}\right) }                     ...(1)

\mathrm{and\, \, \lim _{x \rightarrow 0} \sec ^{-1}\left(\frac{\sin x}{x}\right) }                            ...(2)

Option: 1

(1) exists, (2) does not exist
 


Option: 2

(1) does not exist, (2) exists
 


Option: 3

both (1) and (2) exist
 


Option: 4

neither (1) nor (2) exists


Answers (1)

best_answer

\mathrm{\frac{x}{\sin x}}  is more than 1 in the neighbourhood of ' 0 '.

Hence \mathrm{\lim _{x \rightarrow 0}^{\sec ^{-1}\left(\frac{x}{\sin x}\right)}} exists

while \mathrm{\frac{\sin x}{x}} is less than 1 in the neighbourhood of ' 0 '.

Hence \mathrm{\lim _{x \rightarrow 0} \sec ^{-1}\left(\frac{\sin x}{x}\right)} does not exist.

Posted by

chirag

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