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It is given that f′ (a) exists, then lim...

It is given that f′ (a) exists, then `lim_(x->a)(xf(a)-af(x))/(x-a)`is equal to:

A

`f(a)-af'(a)`

B

`f'(a)`

C

`-f'(a)`

D

`f(a)+a f'(a)`

Text Solution

Verified by Experts

The correct Answer is:
A
It is given that f'(a) exists.
`therefore underset(x to a)lim (f(x)-f(a))/(x-a)=f'(a)`
Now,
`underset(x to a)lim (x f(a)-a f(x))/(x-a)`
`underset(x to a)lim ((x-a) f(a)-a(f(x)-f(a)))/(x-a)`
`underset(x to a)lim f(a)-a{(f(x)-f(a))/(x-a)}`
`=f(a)-a underset(x to a)lim (f(x)-f(a))/(x-a)=f(a)-a f'(a)`
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