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Let $f(a)=g(a)=k$ and their $n^{th}$ derivatives $f^n(a)$, $g^n(a)$ exist and are not equal for some n. Further if $\lim\limits_{x\to a}\large\frac{f(a)g(x)-f(a)-g(a)f(x)+f(a)}{g(x)-f(x)}$$=4$, then the value of k is

$(a)\;0\qquad(b)\;4\qquad(c)\;2\qquad(d)\;1$

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$\lim\limits_{x\to a}\large\frac{f(a)g'(x)-g(a)f'(x)}{g'(x)-f'(x)}$$=4$
$\lim\limits_{x\to a}\large\frac{kg'(x)-kf'(x)}{g'(x)-f'(x)}$$=k$
$\Rightarrow k=4$
Hence (b) is the correct answer.
answered Dec 31, 2013 by sreemathi.v
 

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