Use l’Hôpital’s rule to find the limits in Exercises 7–52.
47. lim (t → ∞) (e^t + t²) / (e^t - t)

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Identify the limit expression: \(\lim_{t \to \infty} \frac{e^{t} + t^{2}}{e^{t} - t}\).

Check the form of the limit by analyzing the behavior of numerator and denominator as \(t \to \infty\). Both numerator and denominator tend to infinity, so the limit is of the form \(\frac{\infty}{\infty}\), which is an indeterminate form suitable for l'Hôpital's Rule.

Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(t\): differentiate numerator \(\frac{d}{dt}(e^{t} + t^{2}) = e^{t} + 2t\), and denominator \(\frac{d}{dt}(e^{t} - t) = e^{t} - 1\).

Rewrite the limit using the derivatives: \(\lim_{t \to \infty} \frac{e^{t} + 2t}{e^{t} - 1}\).

Evaluate the new limit by considering the dominant terms as \(t \to \infty\) and determine the behavior of the fraction to find the limit.

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주요 개념

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

l’Hôpital’s Rule

l’Hôpital’s Rule is a method used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met.

Limits at Infinity

Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how exponential and polynomial functions behave as the variable approaches infinity is crucial for comparing growth rates and determining the limit.

Growth Rates of Exponential vs Polynomial Functions

Exponential functions grow faster than any polynomial function as the variable approaches infinity. Recognizing this helps simplify limits by identifying dominant terms, which is essential when applying l’Hôpital’s Rule or comparing terms in the numerator and denominator.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.47. lim (t → ∞) (e^t + t²) / (e^t - t)