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If $f(x)=\sqrt{\frac{x-\sin x}{x+\cos ^{2} x}}$, then $\lim _{x \rightarrow \infty} f(x)$ is
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1
$\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow \infty} \sqrt{\frac{x-\sin x}{x+\cos ^{2} x}}$
$=\lim _{x \rightarrow \infty} \sqrt{\frac{1-\frac{\sin x}{x}}{1+\frac{\cos ^{2} x}{x}}}$
$=\sqrt{\frac{1-0}{1+0}}$
$\left[\because \frac{\sin x}{x} \rightarrow 0, \frac{\cos ^{2} x}{x} \rightarrow 0\right.$ as $\left.x \rightarrow \infty\right]$
$=1$
$=\lim _{x \rightarrow \infty} \sqrt{\frac{1-\frac{\sin x}{x}}{1+\frac{\cos ^{2} x}{x}}}$
$=\sqrt{\frac{1-0}{1+0}}$
$\left[\because \frac{\sin x}{x} \rightarrow 0, \frac{\cos ^{2} x}{x} \rightarrow 0\right.$ as $\left.x \rightarrow \infty\right]$
$=1$
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