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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 `lim_(xtoa) (f(a)g(x)-f(a)-g(a)f(x)+g(a))/(g(x)-f(x))=4` then find the value of k.

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Verified by Experts

`underset(xtoa)lim(f(a)g(x)-f(a)-g(a)f(x)+g(a))/(g(x)-f(x))=4`
Clearly the LHS is of 0/0 form.
`:." "underset(xtoa)lim(f(a)g'(x)-f(a)-g(a)f'(x))/(g'(x)-f'(x))=4" "`(Using L'Hospital's rule)
`or" "underset(xtoa)lim(f(a)g'(a)-g(a)f'(a))/(g'(a)-f'(a))=4`
or `" "(kg'(a)-kf'(a))/(g'(a)-f'(a))=4`
or k=4
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