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Chapter 1: Problem 6

$\lim _{x \rightarrow \infty} x^{2}\left[\tan ^{-1} \frac{2 x^{2}+1}{x^{2}+2}-\tan ^{-1} 2\right]$ is (A) \(\frac{3}{5}\) (B) \(-\frac{3}{5}\) (C) \(\frac{5}{3}\) (D) \(-\frac{5}{3}\)

Short Answer

Expert verified
Question: Determine the limit of the following expression: \(\lim_{x \rightarrow \infty} x^2\left[\tan^{-1} \frac{2x^2+1}{x^2+2} - \tan^{-1} 2\right]\). Answer: The limit does not converge.

Step by step solution

01

Rewriting the expression inside the arctangent function

In order to simplify the expression, we rewrite the expression inside the first arctangent function as follows: $$ \tan ^{-1} \frac{2 x^{2}+1}{x^{2}+2} = \tan ^{-1} \frac{2x^2+1 - (x^2+2)}{x^2+2} = \tan ^{-1} \frac{x^2-1}{x^2+2} $$ So our expression becomes: $$ \lim _{x \rightarrow \infty} x^{2}\left[\tan ^{-1} \frac{x^2-1}{x^2+2} - \tan ^{-1} 2\right] $$
02

Applying a trigonometric identity

We use the identity \(\tan^{-1}a - \tan^{-1}b = \tan^{-1}\frac{a-b}{1+ab}\) to rewrite the difference between arctangent functions: $$ \lim _{x \rightarrow \infty} x^{2}\left[\tan ^{-1} \frac{\frac{x^2-1}{x^2+2} - 2}{1+\frac{x^2-1}{x^2+2}\cdot2}\right] $$
03

Simplifying the expression

We simplify the expression inside the limit: $$ \lim _{x \rightarrow \infty} x^{2}\left[\tan ^{-1} \frac{(x^2-1)-2(x^2+2)}{1+2(x^2-1)}\right] $$ $$ \lim _{x \rightarrow \infty} x^{2}\left[\tan ^{-1} \frac{-x^2-5}{x^2-2}\right] $$ Now, dividing both the numerator and denominator by \(x^2\), we get: $$ \lim _{x \rightarrow \infty} x^{2}\left[\tan ^{-1} \frac{-1-\frac{5}{x^2}}{1-\frac{2}{x^2}}\right] $$
04

Applying properties of limits

As \(x \rightarrow \infty\), \(\frac{5}{x^2}\) and \(\frac{2}{x^2}\) both approach 0. Therefore, we get: $$ \lim _{x \rightarrow \infty} x^{2}\left[\tan ^{-1} \frac{-1}{1}\right] = x^2 \tan^{-1}(-1) $$ Since \(\tan^{-1}(-1) = -\frac{\pi}{4}\) (arctangent of -1 is \(-\frac{\pi}{4}\)), we get: $$ \lim _{x \rightarrow \infty} x^2 \cdot \left(-\frac{\pi}{4}\right) = -\frac{\pi}{4} \lim _{x \rightarrow \infty} x^2 $$ As x approaches infinity, the limit \(-\frac{\pi}{4} \lim _{x \rightarrow \infty} x^2\) becomes infinite. Therefore, the limit does not converge to any of the given answer choices (A, B, C, D).

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