Use a graph of f to estimate lim⁡x→af(x){\displaystyle\lim_{x\to a}}f\left(x\right) or to show that the limit does not exist. Evaluate f(x) near x=ax=a to support your conjecture.f(x)=1−cos⁡(2x−2)(x−1)2;a=1f\left(x\right)=\frac{1-\cos\left(2x-2\right)}{\left(x-1\$$right)^2$$};a=1

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

문제 2.2.29

2장, 문제 2.2.29

Use a graph of f to estimate $\lim_{x \to a} f (x)$ or to show that the limit does not exist. Evaluate f(x) near $x = a$ to support your conjecture.
$f (x) = \frac{1 - \cos (2 x - 2)}{{(x - 1)}^{2}}; a = 1$

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Identify the function given: $ f(x) = \frac{1 - \cos(2x - 2)}{(x - 1)^2} $ and the point $ a = 1 $.

Recognize that the limit $ \lim_{x \to 1} f(x) $ involves a $ \frac{0}{0} $ indeterminate form, as both the numerator and denominator approach zero when $ x = 1 $.

Apply L'Hôpital's Rule, which is used to evaluate limits of indeterminate forms $ \frac{0}{0} $ or $ \frac{\infty}{\infty} $, by differentiating the numerator and the denominator separately.

Differentiate the numerator: $ \frac{d}{dx}[1 - \cos(2x - 2)] = 2\sin(2x - 2) $.

Differentiate the denominator: $ \frac{d}{dx}[(x - 1)^2] = 2(x - 1) $.

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주요 개념

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Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a certain value. It is denoted as lim(x→a) f(x) and indicates what value f(x) approaches as x gets closer to a. Understanding limits is crucial for analyzing continuity and differentiability, as well as for evaluating functions that may not be defined at certain points.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For the function f(x) to be continuous at x = a, it must satisfy three conditions: f(a) must be defined, the limit as x approaches a must exist, and both must be equal. Discontinuities can lead to limits that do not exist or are undefined.

Graphical Interpretation of Limits

Using a graph to estimate limits involves observing the behavior of the function as it approaches a specific x-value. By analyzing the graph, one can identify trends, such as whether the function approaches a finite value, diverges, or oscillates. This visual approach aids in understanding the concept of limits and can provide insights into the existence or non-existence of limits at certain points.