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If $f(a)=2$, $f'(a)=1$, $g(a)=-1$, $g'(a)=2$, then the value of $\lim\limits_{x\to a}\large\frac{g(x)f(a)-g(a)f(x)}{x-a}$ is

$(a)\;-5\qquad(b)\;1/5\qquad(c)\;5\qquad(d)\;None\;of\;these$

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$\lim\limits_{x\to a}\large\frac{g(x)f(a)-g(a)f(x)}{x-a}$
$\Rightarrow \lim\limits_{h\to 0}\large\frac{g(a+h)f(a)-g(a)f(a+b)}{h}$
$\Rightarrow \lim\limits_{h\to 0}\large\frac{g(a+h)f(a)-g(a)f(a)+g(a)f(a)-g(a)f(a+b)}{h}$
$\Rightarrow \lim\limits_{h\to 0}f(a)\bigg[\large\frac{g(a+h)-g(a)}{h}\bigg]-$$\lim\limits_{h\to 0}g(a)\bigg[\large\frac{f(a+h)-f(a)}{h}\bigg]$
$\Rightarrow f(a)g'(a)-g(a)f'(a)$
$\Rightarrow 2\times 2-(-1)\times 1$
$\Rightarrow 5$
Hence (c) is the correct answer.
answered Dec 31, 2013 by sreemathi.v
 

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