Compare the growth rates of {n¹⁰⁰} and {eⁿ⁄¹⁰⁰} as n → ∞.

검증된 단계별 안내

Identify the two functions to compare: \(f(n) = n^{100}\) and \(g(n) = e^{\frac{n}{100}}\).

Recall that to compare growth rates as \(n \to \infty\), we often consider the limit of their ratio, such as \(\lim_{n \to \infty} \frac{f(n)}{g(n)}\) or \(\lim_{n \to \infty} \frac{g(n)}{f(n)}\).

Set up the limit \(L = \lim_{n \to \infty} \frac{n^{100}}{e^{\frac{n}{100}}}\) to analyze which function grows faster.

Apply L'Hôpital's Rule if the limit is an indeterminate form like \(\frac{\infty}{\infty}\) by differentiating numerator and denominator with respect to \(n\) repeatedly, or use properties of exponential and polynomial functions to reason about the limit.

Conclude the comparison based on the limit: if \(L = 0\), then \(g(n)\) grows faster; if \(L = \infty\), then \(f(n)\) grows faster; if \(L\) is a finite nonzero constant, they grow at comparable rates.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Growth

Exponential growth refers to functions where the variable appears in the exponent, such as e^(n/100). These functions increase much faster than polynomial functions as n becomes very large, often dominating growth rates in limits approaching infinity.

Polynomial Growth

Polynomial growth involves functions where the variable is raised to a fixed power, like n^100. Although these functions grow quickly for large n, their growth rate is slower compared to exponential functions when n approaches infinity.

Limit Comparison of Growth Rates

To compare growth rates as n → ∞, we analyze the limit of the ratio of the two functions. If the limit is zero, the numerator grows slower; if infinite, it grows faster. This method helps determine which function dominates in the long run.