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Evaluate lim(xto pi//2) (sinx-(sinx)^(si...

Evaluate `lim_(xto pi//2) (sinx-(sinx)^(sinx))/(1-sinx+log_(e)sinx)`.

Text Solution

Verified by Experts

Putting `t=sinx`, we have
`underset(t to1)lim(t-t^(t))/(1-t+log_(e)t)" "("form "(0)/(0))`
`=underset(t to1)lim(1-t^(t)(1+log_(e)t))/(0-1+(1)/(t))" "`(Using L'Hospital's rule)
`=underset(t to1)lim(0-{t^(t)((1)/(t))+(1+log_(e)t).t^(t).(1+log_(e)t)})/(0-(1)/(t^(2)))" "`(Using L'Hospital's rule)
`=-(1+1)/(-1)=2`
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