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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)+g(a)}{g(x)-f(x)}$$=4$ then the value of K is

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

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$\lim\limits_{x\to a}\large\frac{f(a)g(x)-g(a)f(x)}{g(x)-f(x)}-$$\lim\limits_{x\to a}\large\frac{f(a)-g(a)}{g(x)-f(x)}$
$\Rightarrow \lim\limits_{x\to a}\large\frac{f(a)(g(x)-g(a))-g(a)(f(x)-f(a))}{g(x)-f(x)}$$-0$
$\Rightarrow K\lim\limits_{x\to a}\large\frac{g(x)-f(x)}{g(x)-f(x)}$$-0=4$(given)
$\Rightarrow K=4$
Hence (d) is the correct answer.
answered Jan 6, 2014 by sreemathi.v
 

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