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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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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Key Concepts

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

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.

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.

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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