For which of the following does lim f(x) = 0 ? x to infinity I. f(x) = lnx / x^99 II. f(x) = e^x / lnx III. f(x) = x^99 / e^x A. I only B. II only C. III only D. I and II only E. I and III only

To determine the limits as \( x \to \infty \) for the given functions, we'll analyze each one individually.

### I. \( f(x) = \frac{\ln x}{x^{99}} \)

As \( x \to \infty \), the numerator \( \ln x \) grows slowly compared to the denominator \( x^{99} \), which grows extremely fast. So, this expression approaches 0. Specifically:

\[
\lim_{x \to \infty} \frac{\ln x}{x^{99}} = 0
\]

Thus, the limit for this function is 0.

### II. \( f(x) = \frac{e^x}{\ln x} \)

As \( x \to \infty \), the exponential function \( e^x \) grows much faster than \( \ln x \). So, this expression tends to infinity:

\[
\lim_{x \to \infty} \frac{e^x}{\ln x} = \infty
\]

Thus, the limit for this function is not 0.

### III. \( f(x) = \frac{x^{99}}{e^x} \)

As \( x \to \infty \), the exponential function \( e^x \) grows significantly faster than any polynomial (like \( x^{99} \)), causing the fraction to approach 0:

\[
\lim_{x \to \infty} \frac{x^{99}}{e^x} = 0
\]

Thus, the limit for this function is 0.

### Conclusion

The functions for which the limit is 0 are I and III. Therefore, the correct answer is:

\[
\boxed{E}
\]

Do you want further details or have any questions?

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Here are 5 related questions:
1. What is the limit of \( \frac{x^{50}}{e^x} \) as \( x \to \infty \)?
2. How does the growth of logarithmic functions compare to polynomials?
3. Can \( \lim_{x \to \infty} \frac{e^x}{x^k} = 0 \) for any constant \( k \)?
4. Why does \( \ln x \) grow slower than \( x \)?
5. What is the behavior of \( \frac{x}{e^x} \) as \( x \to \infty \)?

**Tip:** Exponential functions grow faster than any polynomial or logarithmic function as \( x \to \infty \).

Explore how to solve limit problems involving logarithmic, exponential, and polynomial growth functions as x approaches infinity. Detailed solutions to determine for which functions lim f(x) = 0, including f(x) = ln(x) / x^99, f(x) = e^x / ln(x), and f(x) = x^99 / e^x. Ideal for college-level calculus students.

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