Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (sin 3x) / x

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Theorem 3.10 refers to the limit property: lim x→0 (sin x) / x = 1. This theorem is useful for evaluating limits involving sine functions as x approaches 0.

To apply Theorem 3.10 to the given limit, we need to manipulate the expression lim x→0 (sin 3x) / x to match the form of the theorem.

Notice that the argument of the sine function is 3x, not x. To use the theorem, we can rewrite the limit as lim x→0 (sin 3x) / (3x) * 3.

This manipulation allows us to separate the constant factor 3 from the limit expression, resulting in 3 * lim x→0 (sin 3x) / (3x).

Now, apply Theorem 3.10 to the expression lim x→0 (sin 3x) / (3x), which equals 1. Therefore, the limit becomes 3 * 1.

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Key Concepts

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

Theorem 3.10 (Squeeze Theorem)

Theorem 3.10, often referred to as the Squeeze Theorem, states that if a function is squeezed between two other functions that converge to the same limit at a point, then the squeezed function must also converge to that limit. This theorem is particularly useful for evaluating limits that are difficult to compute directly, especially when dealing with trigonometric functions.

Limit of a Function

The limit of a function describes the behavior of that function as the input approaches a certain value. In calculus, limits are fundamental for defining continuity, derivatives, and integrals. Understanding how to evaluate limits, especially at points where functions may be undefined or indeterminate, is crucial for solving problems in calculus.

Trigonometric Limits

Trigonometric limits involve evaluating the limits of functions that include trigonometric expressions, such as sine and cosine. A key result in calculus is that lim x→0 (sin x)/x = 1, which is often used to simplify expressions involving sine functions. Recognizing and applying this result is essential for solving limits that involve trigonometric functions.

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