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Evaluate underset(xto0)lim((sinx)/(x))^(...

Evaluate `underset(xto0)lim((sinx)/(x))^(((sinx)/(x-sinx))).`

A

`e ^(1)`

B

`e ^(-1)`

C

`e ^(2)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
`lim_(x rarr 0)((sinx)/(x))^(((sinx)/(x))/(1-(sinx)/(x)))`
Put `(sin x)/(x) = t ` , then `t rarr 1 ` when `x rarr 0 " " :. ` limit = `"limt"_(trarr1)^((t)/(1-t))` which is of the form `1^(oo)`
`e^(lim_(t to1) logt^(1/(1-t)))= e^(lim_(t to1) (t)/(1-t)logt)= e^(lim_(t to1) (logt+1)/(-1))` (using L’ Hospital’s rule)
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