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Evaluate (lim(x->oo))(x(log)e{(sin(a+1/...

Evaluate `(lim_(x->oo))(x(log)_e{(sin(a+1/x))/(sina)})`, `0 lt a lt pi/2`

Text Solution

Verified by Experts

`underset(xtooo)limx.log_(e){(sin(a+(1)/(x)))/(sina)}`
`=underset(xtooo)lim(log_(e)((sin(a+(1)/(x)))/(sina)))/((1)/(x))" "`(0/0 form)
`=underset(xtooo)lim((sina)/(sin(a+1//x))xxcos(a+1//x)/(sina)xx(-1(1)/(x^(2))))/(-(1)/(x^(2)))`
(Using L'Hospital's rule)
`=underset(xtooo)limcot(a+(1)/(x))`
`=cota`
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