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Chapter 1: Problem 26

The value of \(\lim _{x \rightarrow \infty}\left(x^{4}(\ln x)^{16}\right)\) is (A) (B) 0 (C) \(\frac{1}{2}\) (D) \(-\frac{1}{2}\)

Short Answer

Expert verified
Answer: The limit of the function as x approaches infinity is infinite.

Step by step solution

01

Analyzing the Behavior of \(x^4\) as \(x\) approaches infinity

As x approaches infinity, \(x^4\) will also tend towards infinity because raising a number to a power increases its value.
02

Analyzing the Behavior of \((\ln x)^{16}\) as \(x\) approaches infinity

When x approaches infinity, the natural logarithm \(\ln x\) also tends towards infinity. And as we raise infinity to the power of 16, it will become an even larger infinity.
03

Combining the Behaviors of \(x^4\) and \((\ln x)^{16}\)

As both \(x^4\) and \((\ln x)^{16}\) tend towards infinity, their product will also tend to infinity. Therefore, we have: $$\lim _{x \rightarrow \infty}\left(x^{4}(\ln x)^{16}\right) = \infty$$ From the given options, there is no answer choice that represents infinity, so the correct option is not specified.

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Most popular questions from this chapter

$\sum_{r=1}^{\infty} \frac{r^{3}+\left(r^{2}+1\right)^{2}}{\left(r^{4}+r^{2}+1\right)\left(r^{2}+r\right)}$ is equal to (A) \(3 / 2\) (B) 1 (C) 2 (D) infinite

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