91Ó°ÊÓ

Theorem II in calculus is a useful tool when dealing with trigonometric limits, especially those involving the sine function. It states that the limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). This simple yet powerful result can be extended to any linear combination of \( x \), such as \( ax \) where \( a \) is a constant, which means \( \lim_{x \to 0} \frac{\sin(ax)}{ax} = 1 \).
This theorem is essential in solving limits that are in the form of a fraction where a sine function is divided by the variable, as illustrated in our exercise:

• We simplify the original expression \( \frac{\sin 3x}{x} \) by rewriting it as \( \frac{\sin 3x}{3x} \cdot 3 \).
• This allows us to match it to Theorem II's form \( \frac{\sin u}{u} \).

The ability to manipulate an expression to fit this form is a crucial skill in calculus, making Theorem II an indispensable part of evaluating trigonometric limits.

Trigonometric limits often involve the sine and cosine functions in scenarios where these functions represent small angle measures. Understanding how to approach them involves recognizing key limits, like \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) as used in Theorem II.
In solving our problem, the manipulation of \( \frac{\sin 3x}{x} \) into \( \frac{\sin 3x}{3x} \cdot 3 \) highlights a systematic approach:

• Identify the part of the expression that represents an angle, like \( 3x \).
• Ensure the denominator matches that angle to make use of trigonometric limit theorems.

Other notable trigonometric limits include those involving the cosine function, such as \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \). These limits form the backbone of evaluating more complex trigonometric expressions.

Limit evaluations are a core component in calculus, providing insight into the behavior of functions as they approach specific points. They help understand how functions behave near critical values, like \( x = 0 \) in our exercise.
Effective evaluation of limits relies on knowing special limits, such as trigonometric limits, and employing strategies to simplify complex expressions.

• Begin by rewriting the expression to match a known limit, applying identities or factoring if needed.
• After simplifications, substitute the point of interest—here \( x \to 0 \)—and use the properties of limits to find the value.

In our problem, the steps led us from the original expression \( \lim_{x \to 0} \frac{\sin 3x}{x} \) to a simplified version, which quickly evaluates to the final result of 3. Mastery of these techniques means easier and faster evaluations for more challenging limits.