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The sine function, denoted as \( \sin x \), is a fundamental concept in trigonometry. It is a periodic, continuous function that describes how the values of angles translate into coordinates on a circle. The sine function oscillates between -1 and 1 and is primarily used to model periodic phenomena.
The sine function is especially important when evaluating limits, as seen in problems involving expressions like \( \lim_{x \rightarrow 0} \frac{\sin x}{x} \). This limit is special because as \(x\) approaches zero, \( \frac{\sin x}{x} \) approaches 1. This property allows us to evaluate more complex limits, especially those involving modifications of sine like \( \sin 3x \).
When solving problems with sine, remember that the behavior around zero is pivotal for analysis. Knowing the outcome of \( \lim_{x \rightarrow 0} \frac{\sin x}{x} \) equips you to manipulate similar terms and derive limits effectively.

Limit properties provide us with tools to simplify and evaluate expressions as the input values approach certain numbers. These properties include rules like the sum, product, and quotient of limits, which help evaluate complex limit expressions.
In particular, the limit property \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \) is widely used in trigonometric limit problems. This property allows substitution when approaching zero, especially helpful when the numerator involves a sine function.
For example, in the original exercise, by recognizing \( \frac{\sin 3x}{3x} \) as a similar form to \( \frac{\sin x}{x} \), we can use these properties. We rewrite the limit to reflect a known property, allowing for easier evaluation. This approach is a powerful strategy in calculus, simplifying evaluations into manageable parts.

Calculating limits involves determining the value that a function approaches as the input approaches a specific point. This process is crucial in understanding the behavior of functions, particularly at points of discontinuity or where direct substitution is not possible.
To calculate the limit of \( \lim_{x \rightarrow 0} \frac{\sin 3x}{5x} \), we first modify the expression to utilize a known limit. We can rewrite the expression as \( \frac{3}{5} \frac{\sin 3x}{3x} \). This is because multiplying and dividing by the same number (3 in this case) does not change the function's value but allows usage of the limit \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \).
Once rewritten, substitute the known limit property \( \frac{\sin 3x}{3x} \) to equal 1 when \( x \rightarrow 0 \). Thus, the final calculation simplifies to \( \frac{3}{5} \times 1 = \frac{3}{5} \). This method highlights how rewriting expressions and applying core limit properties (like those of sine) are key to problem-solving in calculus.