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Evaluate lim(xto0) (x)^((1)/(log(e)sinx)...

Evaluate `lim_(xto0) (x)^((1)/(log_(e)sinx)).`

Text Solution

Verified by Experts

Let `y= underset(xto0^(+))lim(x)^((1)/(log_(e)sinx))" "(0^(0)" form")`
`:." "log_(e)y=log_(e)(underset(xto0^(+))lim(x)^((1)/(log_(e)sinx)))`
`=underset(xto0^(+))limlog_(e)(x)^((1)/(log_(e)sinx))`
`=underset(xto0^(+))lim(1)/(log_(e)sinx)log_(e)x`
`=underset(xto0^(+))lim((1)/(x))/((cosx)/(sinx))" "`(Using L'Hospital Rule)
`=underset(xto0^(+))lim(tanx)/(x)=1`
`:." "y=e`
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