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Pdf. 01 – Subrang Digest January 2011 Free; Paperback: 4.99. Pdf. 02 – Khawateen Digest June 2011 Free; Paperback: 4.99.Q:

Find the value of $\lim\limits_{x\to\infty} x \tan{(\cos^{ -1} x)}.$

Find the value of $$\lim_{x\to\infty} x \tan{(\cos^{ -1} x)}.$$

My attempt: We can write $\tan{(\cos^{ -1} x)}$ as $\frac{\sin(\cos^{ -1} x)}{\cos(\cos^{ -1} x)}.$
I am stuck here. Please give me some hint. Thanks in advance.

A:

Hint:
$$\cos^{ -1}x=\cos^{ -1}\sqrt{1-x^2}=\cos^{ -1}\frac{1-x}{\sqrt{1+x^2}}$$
Can you take it from here?

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