Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
17. lim(x→∞)arcsec(x)

검증된 단계별 안내

Recall the definition of the arcsecant function: \(\mathrm{arcsec}(x)\) is the inverse of the secant function, so \(y = \mathrm{arcsec}(x)\) means \(x = \sec(y)\).

Understand the domain and range of \(\mathrm{arcsec}(x)\). The domain is \(|x| \geq 1\), and the principal range is \([0, \pi]\) excluding \(\frac{\pi}{2}\), where \(\sec(y)\) is defined.

As \(x \to \infty\), consider what happens to \(y\) such that \(x = \sec(y)\). Since \(\sec(y) = \frac{1}{\cos(y)}\), large values of \(x\) correspond to \(\cos(y)\) approaching zero from the positive side.

Identify the angle \(y\) in \([0, \pi]\) where \(\cos(y)\) approaches zero from the positive side. This angle is \(\frac{\pi}{2}\), but note that \(\sec(y)\) is not defined exactly at \(\frac{\pi}{2}\), so we approach it from the left or right.

Conclude that as \(x \to \infty\), \(\mathrm{arcsec}(x)\) approaches \(\frac{\pi}{2}\) from the appropriate side, so the limit is \(\lim_{x \to \infty} \mathrm{arcsec}(x) = \frac{\pi}{2}\).

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.

Limit of a Function as x Approaches Infinity

This concept involves understanding the behavior of a function as the input variable x grows without bound. It helps determine the value that the function approaches, if any, when x becomes very large. Limits at infinity are essential for analyzing end behavior of functions.

Definition and Domain of the Arcsecant Function

The arcsecant function, denoted arcsec(x), is the inverse of the secant function restricted to its principal branch. Its domain is |x| ≥ 1, and it returns an angle whose secant is x. Knowing its domain and range is crucial for evaluating limits involving arcsec.

Behavior of the Secant and Arcsecant Functions at Infinity

As x approaches infinity, the secant function's inverse, arcsec(x), approaches a specific angle. Understanding how sec(θ) behaves and how arcsec(x) relates to θ helps find the limit. Typically, arcsec(x) approaches π/2 as x → ∞ because sec(π/2) is undefined but the limit from the right approaches infinity.

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)17. lim(x→∞)arcsec(x)