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Chapter 1: Problem 33

$\lim _{x \rightarrow-\infty}\left\\{x+\sqrt{x^{2}+3 x \cos \frac{1}{|x|}}\right\\}$ is equal to (A) \(3 / 2\) (B) \(-3 / 2\) (C) \(-1\) (D) none of these

Short Answer

Expert verified
Short Answer: The limit of the given expression as \(x\) approaches \(-\infty\) is \(-\infty\).

Step by step solution

01

Substitute \( -x\) for \( |x| \)

Since \(x\) is approaching to \(-\infty\), \(x\) will always be negative. Therefore, we can replace \(|x|\) with \(-x\) in the given expression. So, we have: $\lim _{x \rightarrow-\infty}\left\\{x+\sqrt{x^{2}+3 x \cos \frac{1}{-x}}\right\\}$
02

Use the identity \(1 - \sin^2\theta = \cos^2\theta\)

Recall that \(1 - \sin^2\theta = \cos^2\theta\). Hence, \(\sin^2\theta = 1 -\cos^2\theta\). Let's first find the value of \(\cos\frac{1}{-x}\) when \(x \rightarrow-\infty\): \(\lim_{x \rightarrow-\infty} \cos\frac{1}{-x} = \cos 0 = 1\) Now, we can rewrite our limit in terms of \(\sin^2\): \(\lim _{x \rightarrow-\infty}\left\\{x+\sqrt{x^{2}+3 x \left(1-\sin^2\frac{1}{-x}\right)}\right\\}\)
03

Factor \(x^2\) and Evaluate the Limit

Next, we factor out \(x^2\) from the square root: \(\lim _{x \rightarrow-\infty}\left\\{x+x \sqrt{1+\frac{3\left(1-\sin^2\frac{1}{-x}\right)}{x}}\right\\}\) As \(x\) goes to \(-\infty\), the term \(\frac{3\left(1-\sin^2\frac{1}{-x}\right)}{x}\) goes to 0, since the maximum value of the numerator is 3. Thus, the expression inside the limit becomes: \(x+x\sqrt{1+0}=x+x=2x\) Now, take the limit: \(\lim _{x \rightarrow-\infty} 2x = -\infty\) So, the correct answer is (D) none of these.

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Most popular questions from this chapter

Assertion (A): $\lim _{x \rightarrow 0}[x]\left(\frac{e^{1 / x}-1}{e^{1 / x}+1}\right)$ where [.] represents greatest integer function does not exist. Reason $(\mathrm{R}): \lim _{\mathrm{x} \rightarrow 0}\left(\frac{\mathrm{e}^{\mathrm{l} / \mathrm{x}}-1}{\mathrm{e}^{1 / \mathrm{x}}+1}\right)$ does not exist.

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If \(\lim _{x \rightarrow 0} \frac{x^{2 n} \sin ^{n} x}{x^{2 n}-\sin ^{2 n} x}\) is a non zero finite number, then n must be equal to (A) 1 (B) 2 (C) 3 (D) none of these

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