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Chapter 1: Problem 23

$\lim _{x \rightarrow-\infty} \frac{x^{5} \tan \left(\frac{1}{\pi x^{2}}\right)+3|x|^{2}+7}{|x|^{3}+7|x|+8}$ is equal to (A) \(\pi\) (B) \(\frac{1}{\pi}\) (C) \(-\frac{1}{\pi}\) (D) None of these

Short Answer

Expert verified
Function: $$\lim_{x \to -\infty} \frac{-x^{5} \tan \left(\frac{1}{-\pi x^{2}}\right)+3x^{2}+7}{x^{3}+7x+8}$$ Answer choices: a) -$\pi /2$ b) $\pi /2$ c) $\pi /4$ d) None of these Solution: After analyzing and simplifying the function, we found out that the limit of the function as x approaches negative infinity is -3. Therefore, the correct answer is (D) None of these.

Step by step solution

01

Rewrite the expression

Since we are dealing with large negative values for x, we can rewrite the function using negative powers of x and the definition of tan(z) = y/x, z = arctan(y/x): $$\lim_{x \to -\infty} \frac{-x^{5} \tan \left(\frac{1}{-\pi x^{2}}\right)+3x^{2}+7}{x^{3}+7x+8}$$ Knowing that tan(z) = y/x, z = arctan(y/x), and the first and second term in the numerator has an x in common. Therefore the function simplifies to: $$\lim_{x \to -\infty} \frac{-x \left(x^{4} \tan \left(\frac{1}{-\pi x^{2}}\right)+3x\right)+7}{x^{3}+7x+8}$$
02

Use algebra to simplify further

As x approaches negative infinity, most of the terms will be dominated by x. We can simplify this expression further by diving the numerator and the denominator by x^3: $$\lim_{x \to -\infty} \frac{- \left(x \tan \left(\frac{1}{-\pi x^{2}}\right)+3\right)+\frac{7}{x^{3}}}{1+\frac{7}{x^{2}}+\frac{8}{x^{3}}}$$
03

Analyze the behavior of the limit

Now, we will examine each term in the simplified expression individually: 1. \(x\tan \left(\frac{1}{-\pi x^{2}}\right)\): As x approaches negative infinity, the argument of the tangent function approaches zero. Therefore, this term will approach zero because \(\tan(0) = 0\). 2. \(\frac{7}{x^{3}}\), \(\frac{7}{x^{2}}\), and \(\frac{8}{x^{3}}\): As x approaches negative infinity, these terms will approach zero since they are all rational functions with higher powers of x in the denominator compared to the numerator. Combining this information, we can rewrite the limit as: $$\lim_{x \to -\infty} \frac{- (0 +3)+0}{1+0+0} = -\frac{3}{1}$$
04

Compare the result with answer choices

The limit of the function as x approaches negative infinity is -3, which does not match any of the given answer choices. Therefore, the correct answer is (D) None of these.

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