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If the function $\mathrm{f}(\mathrm{x})=\frac{2 \mathrm{x}-\sin ^{-1} \mathrm{x}}{2 \mathrm{x}+\tan ^{-1} \mathrm{x}}$ is continuous at each
point in its domain, then what is the value of $\mathrm{f}(0)$ ?
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point in its domain, then what is the value of $\mathrm{f}(0)$ ?
Solution:
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Verified Answer
The correct answer is:
$\frac{1}{3}$
$f(0)=\lim _{x \rightarrow 0} \frac{2 x-\sin ^{-1} x}{2 x+\tan ^{-1} x}$
$=\lim _{x \rightarrow 0} \frac{2-\frac{\sin ^{-1} x}{x}}{2+\frac{\tan ^{-1} x}{x}}=\frac{2-1}{2+1}=\frac{1}{3}$
$=\lim _{x \rightarrow 0} \frac{2-\frac{\sin ^{-1} x}{x}}{2+\frac{\tan ^{-1} x}{x}}=\frac{2-1}{2+1}=\frac{1}{3}$
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