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Question: Let \(f(a) = g(a) = k\) and their \(n^{th}\) derivatives \(f^{n}(a),g^{n}(a)\) exist and are not equ...

Let f(a)=g(a)=kf(a) = g(a) = k and their nthn^{th} derivatives fn(a),gn(a)f^{n}(a),g^{n}(a) exist and are not equal for some n. If

limxaf(a)g(x)f(a)g(a)f(x)+g(a)g(x)f(x)=4,\lim_{x \rightarrow a}\frac{f(a)g(x) - f(a) - g(a)f(x) + g(a)}{g(x) - f(x)} = 4, then the value of k is

A

4

B

2

C

1

D

0

Answer

4

Explanation

Solution

limxakg(x)kf(x)g(x)f(x)=4\lim_{x \rightarrow a}\frac{kg(x) - kf(x)}{g(x) - f(x)} = 4

By L-Hospital’ rule, limxak[g(x)f(x)g(x)f(x)]=4\lim_{x \rightarrow a}k\left\lbrack \frac{g^{'}(x) - f^{'}(x)}{g^{'}(x) - f^{'}(x)} \right\rbrack = 4, ∴ k=4k = 4.