Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.6.17

Chapter 7, Problem 7.6.17

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

Verified step by step guidance

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 \(\lim_{x \to \infty} \mathrm{arcsec}(x) = \frac{\pi}{2}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Graphs of Secant and Cosecant Functions

Related Practice

Textbook Question

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Textbook Question

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Textbook Question

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

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Textbook Question

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