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

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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.

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 \( x \).

To apply Theorem 3.10, we need the expression inside the sine function to match the denominator. Rewrite the limit as \( \lim_{x \to 0} \frac{7}{3} \cdot \frac{\sin 7x}{7x} \).

Recognize that \( \frac{\sin 7x}{7x} \) is in the form required by Theorem 3.10, so \( \lim_{x \to 0} \frac{\sin 7x}{7x} = 1 \).

Combine the results to find the limit: \( \lim_{x \to 0} \frac{7}{3} \cdot 1 = \frac{7}{3} \). Thus, the limit is \( \frac{7}{3} \).

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