110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
f. f(x) = sech(x), g(x) = e^(-x)

검증된 단계별 안내

Recall the definitions of the functions: \(f(x) = \text{sech}(x)\) and \(g(x) = e^{-x}\). The hyperbolic secant function is defined as \(\text{sech}(x) = \frac{2}{e^{x} + e^{-x}}\).

Analyze the behavior of \(f(x)\) as \(x \to \infty\). Since \(e^{x}\) grows very large, the term \(e^{x} + e^{-x}\) is dominated by \(e^{x}\), so \(\text{sech}(x) \approx \frac{2}{e^{x}} = 2e^{-x}\) for large \(x\).

Compare this approximation of \(f(x)\) to \(g(x) = e^{-x}\). Notice that \(f(x)\) behaves like \(2e^{-x}\), which is just a constant multiple of \(g(x)\) as \(x \to \infty\).

Since \(f(x)\) and \(g(x)\) differ only by a constant factor in their dominant terms for large \(x\), they decay at the same exponential rate as \(x \to \infty\).

Conclude that \(f\) and \(g\) grow (or decay) at the same rate as \(x \to \infty\) because their leading terms have the same exponential order.

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 approaches a limit, often infinity. Understanding this helps compare growth rates by analyzing limits of ratios or differences as x→∞, revealing which function dominates or if they grow at similar rates.

Hyperbolic Secant Function (sech)

The hyperbolic secant, sech(x), is defined as 1/cosh(x), where cosh(x) = (e^x + e^{-x})/2. As x→∞, cosh(x) grows exponentially, so sech(x) approaches zero exponentially, behaving similarly to e^{-x} but with a specific constant factor.

Exponential Decay and Comparison

Exponential decay functions like e^{-x} decrease rapidly as x→∞. Comparing f(x) = sech(x) to g(x) = e^{-x} involves examining their rates of decay by considering limits of their ratio, which determines if one decays faster, slower, or at the same rate.

110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.f. f(x) = sech(x), g(x) = e^(-x)