Ch. 7 - Transcendental Functions

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

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.6.13

Chapter 7, Problem 7.6.13

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
13. lim(x → 1⁻)arcsin(x)

Verified step by step guidance

Identify the function and the limit point: We want to find the limit of \(\arcsin(x)\) as \(x\) approaches 1 from the left, which is written as \(\lim_{x \to 1^-} \arcsin(x)\).

Recall the domain and range of the \(\arcsin\) function: The function \(\arcsin(x)\) is defined for \(x\) in the interval \([-1, 1]\), and its range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Understand the behavior near the limit point: Since \(x\) approaches 1 from values less than 1, we are approaching the right endpoint of the domain of \(\arcsin(x)\) from inside the domain.

Use the continuity of \(\arcsin(x)\) on its domain: The \(\arcsin\) function is continuous on \([-1, 1]\), so the limit as \(x\) approaches 1 from the left is simply \(\arcsin(1)\).

Evaluate \(\arcsin(1)\): Recall that \(\arcsin(1)\) is the angle whose sine is 1, which corresponds to \(\frac{\pi}{2}\).

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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

The limit of a function describes the value that the function approaches as the input approaches a specific point. It helps understand the behavior of functions near points where they may not be explicitly defined. In this problem, we consider the limit as x approaches 1 from the left.

One-Sided Limits

One-sided limits examine the behavior of a function as the input approaches a point from only one side—either from the left (denoted x → a⁻) or the right (x → a⁺). This is important when the function behaves differently on either side of the point, as in the limit from the left of x approaching 1.

Arcsine Function Properties

The arcsine function, denoted arcsin(x), is the inverse of the sine function restricted to [-π/2, π/2]. It is continuous and defined for x in [-1, 1], with arcsin(1) = π/2. Understanding its domain and continuity helps evaluate limits involving arcsin near boundary points.

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