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Let f(x+y)=f(x)f(y)andf(x)=1+xg(x)phi(x)...

Let `f(x+y)=f(x)f(y)andf(x)=1+xg(x)phi(x)" where "lim_(x to 0) g(x)=a andlim_(x to 0) phi(x)=b`. Then what is f(x) equal to?

A

`1+anf(x)`

B

`1+ab`

C

ab

D

abf(x)

Text Solution

Verified by Experts

The correct Answer is:
D
`f'(x)=underset(hto0)lim(f(x+h)-f(x))/(h)=underset(hto0)lim(f(x)f(h)-f(x))/(h)`
`=underset(hto0)lim(f(x)[f(h)-1])/(h)=underset(hto0)lim((f(x)hg(h)phi(h)])/(h)`
`=abf(x)`
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