Let f(a)=g(a)=k and their nth derivative...

Let `f(a)=g(a)=k` and their `nth` derivatives exist and be not equal for some n.
If `underset(xtoa)lim(f(a)g(x)-f(a)-g(a)f(x)+g(a))/(g(x)-f(x))=4` then find the value of k.

A

0

B

4

C

2

D

1

Text Solution

Verified by Experts

The correct Answer is:

B

We have,
`lim_(xtoa)(f(a)g(x)-f(a)-g(a)f(x)+g(a))/(g(x)-f(x))=4`
`rArr lim_(xtoa) (f(a)g'(x)=0-g(a)f'(x)+0)/(g'(x)-f'(x))=4`
`rArr (f(a)g'(a)-g(a)f'(a))/(g'(a)-f'(a))=4 ["Using L' Hospital's Rule"]`
` rArr (k{g'(a)-f'(a)})/({g'(a)-f'(a)})=4 [because f(a)=g(a)=k]`
`rArr k=4`.

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