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The value of $f(0)$ so that the function $f(x)=\large\frac{2x-\sin^{-1}x}{2x+\tan^{-1}x}$ is continuous at each point on its domain is

$(a)\;2\qquad(b)\;\large\frac{1}{3}$$\qquad(c)\;\large\frac{2}{3}$$\qquad(d)\;-\large\frac{1}{3}$

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$f(x)=\lim\limits_{x\to 0}\large\frac{2x-\sin^{-1}x}{2x+\tan^{-1}x}$
$\qquad=\lim\limits_{x\to 0}\large\frac{2-\Large\frac{\sin^{-1}x}{x}}{2+\Large\frac{\tan^{-1}x}{x}}$
$\qquad=\large\frac{2-1}{2+1}$
$\qquad=\large\frac{1}{3}$
Since $f(x)$ is continuous $f(0)=\large\frac{1}{3}$
Hence (b) is the correct answer.
answered Dec 24, 2013 by sreemathi.v
 

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