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Chapter 3: Problem 47

The value of \(\lim _{x \rightarrow \infty}\left(x+\sqrt{x^{2}+3 x \sin \left(\frac{1}{|x|}\right)}\right)\) is (a) \(3 / 2\) (b) \(-3 / 2\) (c) \(-1\) (d) 0

Short Answer

Expert verified
The limit as x approaches infinity for the given equation is \(\infty\). None of the provided options (a) \(3 / 2\), (b) \(-3 / 2\), (c) \(-1\), (d) 0) is correct.

Step by step solution

01

Simplify the Square Root

We see that we are dealing with a square root that involves \(x^{2}\) and another function of \(x\). As \(x\) approaches infinity, the \(x^{2}\) term will dominate the other term inside the square root. Therefore, we can simplify the square root to just \(\sqrt{x^{2}}\). This simplifies our equation to \(x+\vert x \vert\).
02

Simplify Further

Since \(x\) is approaching infinity, we can consider \(x\) to be always positive. Therefore, \(\vert x \vert\) can be replaced with \(x\). The equation further simplifies to \(x+x\).
03

Final Calculation

The simplified equation becomes \(2x\). Now, taking the limit as \(x\) approaches infinity for \(2x\) gives us \(\infty\).

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