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Let f:RrarrR be such that f(a)=1, f(a)=2...

Let `f:RrarrR` be such that `f(a)=1, f(a)=2`. Then `lim_(x to 0)((f^(2)(a+x))/(f(a)))^(1//x)` is

A

`e^(2)`

B

`e^(4)`

C

`e^(-4)`

D

`1//e`

Text Solution

Verified by Experts

The correct Answer is:
B
`underset(xrarr0)(lim)((f^(2)(a+x))/(f(a)))^(1//x)`
`" "=e^(underset(xrarr0)(lim)((f^(2)(a+x)-f(a))/(xf(a))))`
`" "=e^(underset(xrarr0)(lim)(f^(2)(a+x)-(f(a))^(2))/(x))`
`" "=e^(underset(xrarr0)(lim)((f(a+x)-f(a))(f(a+x)+f(a)))/(x))`
`" "=e^(underset(xrarr0)(lim)2f(a)((f(a+x)-f(a)))/(x))`
`" "=e^(underset(xrarr0)(lim)2f(a)f'(a))`
`" "=e^(4)`
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