Exercises 5–10 refer to the functionf(x) = { x² − 1, −1 ≤ x b. Does lim x → −1⁺ f (x) exist?

Ch. 2 - Limits and Continuity

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Hass 15th Edition

Ch. 2 - Limits and Continuity

문제 2.5.5b

2장, 문제 2.5.5b

Exercises 5–10 refer to the function
f(x) = { x² − 1, −1 ≤ x < 0
2x, 0 < x < 1
1, x = 1
−2x + 4, 1 < x < 2
0, 2 < x < 3
graphed in the accompanying figure.
<IMAGE>
b. Does lim x → −1⁺ f (x) exist?

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Identify the piece of the piecewise function that is relevant for x approaching -1 from the right. Since we are interested in the limit as x approaches -1 from the right (x → -1⁺), we need to consider the interval that includes values just greater than -1.

Examine the piece of the function that applies to the interval -1 ≤ x < 0. In this interval, the function is defined as f(x) = x² - 1.

To find the limit as x approaches -1 from the right, evaluate the expression x² - 1 as x gets closer to -1 from values greater than -1.

Substitute values that are slightly greater than -1 into the expression x² - 1 to observe the behavior of the function. For example, consider values like -0.9, -0.99, etc., and calculate the corresponding f(x) values.

Conclude whether the limit exists by determining if the values of f(x) approach a specific number as x approaches -1 from the right. If they do, the limit exists and is equal to that number.

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주요 개념

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps determine the value that a function approaches as the input gets arbitrarily close to a specific point, which is crucial for understanding continuity and differentiability.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. Understanding how to evaluate piecewise functions is essential for analyzing their behavior at specific points, especially at boundaries where the definition of the function changes.

Right-Hand Limit

The right-hand limit of a function at a point is the value that the function approaches as the input approaches that point from the right. This concept is particularly important when evaluating limits at points where the function's definition changes, as it helps determine if the limit exists and what value it approaches.

Exercises 5–10 refer to the functionf(x) = { x² − 1, −1 ≤ x < 02x, 0 < x < 11, x = 1−2x + 4, 1 < x < 20, 2 < x < 3graphed in the accompanying figure.<IMAGE>b. Does lim x → −1⁺ f (x) exist?