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Evaluate: lim(x->0)(cosx)^(cotx)...

Evaluate: `lim_(x->0)(cosx)^(cotx)`

Text Solution

Verified by Experts

`underset(xto0)lim(cosx)^(cotx)=underset(xto0)lim[(1+(cosx-1))^((1)/(cosx-1))]^((cos-1)/(tanx))`
`underset(xto0)lim[(1+(cosx-1))^((1)/(cosx-1))]^(underset(xto0)lim(cos-1)/(tanx))`
`=e^(underset(xto0)lim(cosx-1)/(sinx)cosx`
`=e^(underset(xto0)lim(cosx-1)/(sin^(2)x)cosxsinx`
`=e^(underset(xto0)lim(cosx-1)/(1-cos^(2)x)cosxsins`
`=e^(underset(xto0)lim-(sinxcosx)/(1+cosx)=e^(0)=1`
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