Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (sin 7x) / 3x

Verified step by step guidance

1. Identify Theorem 3.10, which is the standard limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). This theorem is useful for evaluating limits involving sine functions as \( x \) approaches zero.
2. Rewrite the given limit \( \lim_{x \to 0} \frac{\sin 7x}{3x} \) in a form that allows the use of Theorem 3.10. Notice that the argument of the sine function is \( 7x \), not \( x \).
3. To apply Theorem 3.10, we need the expression inside the sine function to match the denominator. Rewrite the limit as \( \lim_{x \to 0} \frac{7}{3} \cdot \frac{\sin 7x}{7x} \).
4. Recognize that \( \frac{\sin 7x}{7x} \) is in the form required by Theorem 3.10, so \( \lim_{x \to 0} \frac{\sin 7x}{7x} = 1 \).
5. Combine the results to find the limit: \( \lim_{x \to 0} \frac{7}{3} \cdot 1 = \frac{7}{3} \). Thus, the limit is \( \frac{7}{3} \).

Key Concepts

Theorem 3.10 (Limit of Sin Function)

Theorem 3.10 typically refers to the limit property that states lim (x→0) (sin(kx)/x) = k for any constant k. This theorem is crucial for evaluating limits involving sine functions, as it provides a straightforward way to simplify expressions as x approaches zero.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus that involves determining the value that a function approaches as the input approaches a certain point. In this case, we are interested in the behavior of the function (sin 7x)/(3x) as x approaches 0, which requires applying limit properties and potentially L'Hôpital's Rule if the limit results in an indeterminate form.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞. In the given limit, substituting x = 0 results in the form 0/0, which necessitates further analysis using limit theorems or algebraic manipulation to resolve the limit correctly.