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lim(xto1)(sin(e^(x-1)-1))/(log x) is equ...

`lim_(xto1)(sin(e^(x-1)-1))/(log x)` is equal to

A

0

B

e

C

`(1)/(e)`

D

`1 `

Text Solution

Verified by Experts

The correct Answer is:
D
`lim_(xrarr1)(sin(e^(x-1)-1))/(logx)=lim_(hrarr0)(sin(e^(h)-1))/(log(1+h))` [Putting x - 1 = h]
`=lim_(xrarr0)(sin(e^(h)-1))/(e^(h)-1).(e^(h)-1)/(h).(h)/(log(1+h))`
`=lim_(xrarr0)(sin(e^(h)-1))/(e^(h)-1).(e^(h)-1)/(h).lim_(hrarr0)(h)/(log(1+h))=1.1.1=1`
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