91Ó°ÊÓ

Trigonometric identities are formulas involving trigonometric functions that are true for all values of the variables where they are defined. When handling calculus problems involving trigonometric functions, these identities are essential tools to simplify expressions or rewrite terms. In the exercise above, the Pythagorean Trigonometric Identity \[\sin^2 x + \cos^2 x = 1\]was used. We can manipulate this identity to express either \(\sin^2 x\) or \(\cos^2 x\) in terms of the other, which is highly useful for solving limits. Here, it was rewritten as:

• \(\sin^2 x = 1 - \cos^2 x\)
• \(1 - \cos^2 x = \sin^2 x\)

This allowed us to replace the \(1 - \cos^2 x\) in the original function with \(\sin^2 x\), making the following steps much more straightforward.

Evaluating limits is a fundamental part of calculus that involves finding the value that a function approaches as the input approaches a certain point. Limits help us understand behaviors in situations involving continuity, rates of change, and more.In this exercise, after applying the trigonometric identity, the function became \[\lim_{x \rightarrow 0^{+}} \frac{\sin^{2} x}{\sin x}\]To evaluate it, we simplify it first (as it will be covered in detail in the next section). The limit can be directly calculated if we understand that as \(x\) approaches 0 from the right, or \(x \rightarrow 0^{+}\), both \(\sin^2 x\) and \(\sin x\) approach 0. Hence,\[\lim_{x \rightarrow 0^{+}} \sin x = 0\]Thus, the entire expression tends towards 0. Understanding these small complexities with directionality (like with \(x \rightarrow 0^{+}\)) is crucial since limits can change with directions.

Simplification in calculus refers to reducing complex expressions into simpler forms. This is highly valuable as simpler equations are easier to work with and understand. In solving the limit provided, simplification played a vital role.Initially, the function was transformed using the identity, simplifying it to the expression:\[\frac{\sin^{2} x}{\sin x}\]This expression can be further simplified by canceling a common factor in the numerator and denominator. Since \(\sin x\) appears in both, it can be cancelled out giving us:\[\lim_{x \rightarrow 0^{+}} \sin x\]This simplification step is crucial because it moves us closer to the limit that can be directly computed. Simplicity not only clarifies the computational steps but also enhances conceptual understanding, making the overall problem more approachable.