Question

Evaluate each limit. $$ \lim _{\theta \rightarrow 0} \frac{\sin ^{2} \theta}{\theta^{2}} $$

Short answer

The limit evaluates to 1.

Step-by-step solution

Understand the Limit Expression

We need to evaluate the limit \( \lim _{\theta \rightarrow 0} \frac{\sin ^2 \theta}{\theta^2} \). This expression involves the function \( \sin(\theta) \) squared divided by \( \theta^2 \), and we are looking to see what value it approaches as \( \theta \) gets closer to 0.

Use Trigonometric Limits

Recall the fundamental trigonometric limit \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \). This is essential for evaluating our given expression.

Rewrite the Limit Expression

Re-write the expression to highlight the square of the trigonometric function:\[ \lim_{\theta \to 0} \frac{\sin^2 \theta}{\theta^2} = \left( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} \right)^2 \]

Apply the Fundamental Limit

Utilizing the fundamental limit from Step 2 on the rewritten expression from Step 3, we have:\[ \left( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} \right)^2 = 1^2 = 1 \]

Evaluate the Limit

Combine the previous steps to conclude that \( \lim _{\theta \rightarrow 0} \frac{\sin ^2 \theta}{\theta^2} = 1 \). Thus, the limit evaluates to 1.

Key concepts

Trigonometric Limits

Trigonometric limits are a crucial aspect of calculus that help us understand the behavior of trigonometric functions as they approach specific values. Understanding these limits is essential for solving a wide variety of calculus problems, especially those that involve sine, cosine, and other periodic functions. A fundamental trigonometric limit to remember is \[\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\]This particular limit is often used when evaluating expressions where direct substitution might lead to an indeterminate form such as \(\frac{0}{0}\).

• Limit properties allow for the manipulation and evaluation of expressions in a more straightforward manner.
• By using established trigonometric limits, it's possible to solve complex limit problems.
• These limits are foundational in understanding continuity and differentiability of trigonometric functions.

Within calculus, recognizing these limits helps simplify the expressions and ensure accurate results.

Sine Function

The sine function \(\sin(\theta)\) is one of the core trigonometric functions, and its behavior as it approaches certain values, like zero, is of significant importance in calculus. The sine function is an odd function, periodic, and oscillates between -1 and 1. This oscillatory nature is particularly telling when evaluating limits like \[\lim _{\theta \rightarrow 0} \frac{\sin ^2 \theta}{\theta^2}\]The sine function is defined based on the unit circle, and its values repeat every \(2\pi\) radians. As \(\theta\) approaches zero, the sine function's behavior is predictable and allows us to make educated evaluations about limits.

• As \(\theta\) moves towards zero, \(\sin(\theta)\) approaches zero at a rate that is proportional to \(\theta\).
• This proportional relationship is crucial for understanding why \(\frac{\sin \theta}{\theta} \to 1\) as \(\theta \to 0\).
• When squared, \(\sin^2(\theta)\) must be carefully handled to avoid mistakes in limit calculations.

This understanding allows us to apply mathematical techniques efficiently in different context.

Evaluation of Limits

The evaluation of limits is a fundamental procedure in calculus used to determine the behavior of functions as they approach a particular point. When evaluating limits involving trigonometric functions like \[\lim_{\theta \to 0} \frac{\sin^2 \theta}{\theta^2},\]we rely on known fundamental limits and algebraic manipulation. Here's how to approach it:

• First, identify if the expression results in an indeterminate form, such as \(\frac{0}{0}\).
• Next, use trigonometric identities or algebraic manipulations to simplify the expression.
• Apply known limits, like \(\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\), to further simplify and resolve the limit.

In our example, by expressing \(\frac{\sin^2 \theta}{\theta^2}\) as \(\left(\frac{\sin \theta}{\theta}\right)^2\), we apply the limit of \(\frac{\sin \theta}{\theta}\) twice. Understanding these steps and the reasoning behind them enables us to evaluate limits accurately, even when faced with complex trigonometric functions.