Question

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

Short answer

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

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: $$\lim_{x \to -\infty} \frac{-x \left(x^{4} \tan \left(\frac{1}{-\pi x^{2}}\right)+3x\right)+7}{x^{3}+7x+8}$$

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: $$\lim_{x \to -\infty} \frac{- \left(x \tan \left(\frac{1}{-\pi x^{2}}\right)+3\right)+\frac{7}{x^{3}}}{1+\frac{7}{x^{2}}+\frac{8}{x^{3}}}$$

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: $$\lim_{x \to -\infty} \frac{- (0 +3)+0}{1+0+0} = -\frac{3}{1}$$

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.