Limits, Continuity & Differentiability
Limits at Infinity
Grade 12
Question:
<p>\(\lim_{x \to \infty} \left[(x+5)\tan^{-1}(x+5) - (x+1)\tan^{-1}(x+1)\right]\) is equal to</p>
<p>(a) \(\frac{\pi}{2}\)</p>
<p>(b) \(2\pi\)</p>
<p>(c) \(\frac{\pi}{4}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the standard limit $\lim_{x \to \infty} x\tan^{-1}(x) = \frac{\pi}{2}$ and properties of inverse trigonometric functions to evaluate the difference.
<p><strong>Solution:</strong></p><p>$\lim_{x \to \infty} \left[(x+5)\tan^{-1}(x+5) - (x+1)\cot^{-1}(x+5) - (x+1)\cot^{-1}(x+1)\right]$</p><p>Using the identity that $\lim_{x \to \infty} x\tan^{-1}(x) = \frac{\pi}{2}$ and properties of inverse trigonometric functions:</p><p>$\lim_{x \to \infty} \left[\frac{2\sin(\pi/2)}{\pi/2}\right] = \frac{\pi}{2}$</p><p>∴ Answer is (a) $\frac{\pi}{2}$</p>
Correct Answer: A