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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 2 - Limits and ContinuityProblema 2.4.35
Capitolo 2, Problema 2.4.35

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limt→0 sin(1 − cos t) / (1 − cos t)

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1
Recognize that the limit involves the expression sin(1 - cos t) / (1 - cos t) as t approaches 0. This is a form that can be related to the standard limit lim(θ→0) sin(θ)/θ = 1.
To use the standard limit, we need to manipulate the expression to match the form sin(θ)/θ. Notice that both the numerator and the denominator have the same expression (1 - cos t).
Apply the standard limit by setting θ = 1 - cos t. As t approaches 0, cos t approaches 1, making θ approach 0.
Rewrite the limit in terms of θ: lim(t→0) sin(1 - cos t) / (1 - cos t) becomes lim(θ→0) sin(θ)/θ.
Since lim(θ→0) sin(θ)/θ = 1, conclude that the original limit also equals 1.

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Limit Definition

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, particularly where they may not be explicitly defined. The notation lim x→a f(x) indicates the value that f(x) approaches as x gets closer to a.
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Definition of the Definite Integral

Trigonometric Limits

Trigonometric limits, such as lim θ→0 sin θ / θ = 1, are specific cases that often arise in calculus. This particular limit is crucial for evaluating limits involving sine and cosine functions, especially when they appear in indeterminate forms. Understanding this limit allows for simplification of expressions involving trigonometric functions as they approach zero.
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L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms like 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when dealing with limits involving trigonometric functions and can simplify complex limit problems.
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Finding Limits


In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.



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At what points are the functions in Exercises 13–30 continuous?


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Limits and Infinity


Find the limits in Exercises 37–46.


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At what points are the functions in Exercises 13–30 continuous?

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Limits of Average Rates of Change


Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.


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Limits with trigonometric functions


Find the limits in Exercises 43–50.


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