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"If "f(x)=lim(hrarr0) ((sin(x+h))^(log(e...

`"If "f(x)=lim_(hrarr0) ((sin(x+h))^(log_(e)(x+h))-(sin x)^(log_(e)x))/(h)" then find "f(pi//2).`

Text Solution

Verified by Experts

`"Let "g(x)=(sin x)^(log_(e^(x)))`
`therefore" "f(x)=underset(hrarr0)lim(g(x+h)-g(x))/(h)=g'(x)`
`therefore" "f(x)=g'(x)=(d)/(dx)(e^(log, x. log_(e) sin x))`
`(sin x)^(log_(e)x)((1)/(x)cdotlog_(e) sin x + log_(e) x.(1)/(sin x)cdot cos x)`
`therefore" "f(pi//2)=0`
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