109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
b. f(x)=x, g(x)=x + 1/x

검증된 단계별 안내

Identify the given functions: \(f(x) = x\) and \(g(x) = x + \frac{1}{x}\).

As \(x \to \infty\), analyze the behavior of each function. For \(f(x) = x\), the function grows without bound linearly.

For \(g(x) = x + \frac{1}{x}\), note that \(\frac{1}{x} \to 0\) as \(x \to \infty\), so \(g(x)\) behaves like \(x\) plus a very small positive term.

Compare the growth rates by considering the ratio \(\frac{f(x)}{g(x)} = \frac{x}{x + \frac{1}{x}}\). Simplify this expression to understand the limit as \(x \to \infty\).

Evaluate the limit \(\lim_{x \to \infty} \frac{f(x)}{g(x)}\) to determine if it approaches 0, a finite nonzero constant, or infinity, which tells us if \(f\) grows slower, at the same rate, or faster than \(g\) respectively.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Asymptotic Behavior of Functions

Asymptotic behavior describes how functions behave as the input grows very large (x→∞). It helps compare growth rates by focusing on dominant terms and ignoring smaller ones that become insignificant at infinity.

Limits and Limit Comparison

Limits evaluate the value a function approaches as x approaches infinity. Comparing the limit of the ratio f(x)/g(x) as x→∞ determines if one function grows faster, slower, or at the same rate as the other.

Dominant Terms in Functions

The dominant term in a function is the part that grows fastest as x→∞. For example, in g(x) = x + 1/x, the term x dominates because 1/x approaches zero, so the growth rate is mainly determined by x.

109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.b. f(x)=x, g(x)=x + 1/x