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Let f(x^m y^n)=mf(x)+nf(y) for all x , y...

Let `f(x^m y^n)=mf(x)+nf(y)` for all `x , y in R^+` and for all `m ,n in Rdot` If `f^(prime)(x)` exists and has the value `e/x ,` then find `(lim)_(xvec0)(f(1+x))/x`

Text Solution

Verified by Experts

The correct Answer is:
e
For any `x in R^(+),` we have
`therefore" "f(1)=f(1)+f(1)" [Putting x = y = m= n =1]"`
`"or "f(1)=0`
`"or "underset(xrarr0)lim(f(1+x))/(x)=underset(xrarr0)lim(f'(1+x))/(1)" (using L' Hopital's rule)"`
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