Problem 9 - Calculus - AP Edition

Question

Find the limits. $$\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta}$$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understand the Problem

We need to find the limit of \( \frac{\tan \theta}{\theta} \) as \( \theta \) approaches 0.

Step 2: Recall Trigonometric Limits

Recall the special trigonometric limit \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \). This is a basic limit identity that is often used in calculus.

Step 3: Use the Sine and Cosine Identity for Tangent

Express \( \tan \theta \) in terms of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substituting this into the limit gives us \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta \cos \theta} \).

Step 4: Separate the Expressions

Consider the limit \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta \cos \theta} \), which can be split as \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} \times \lim_{\theta \to 0} \frac{1}{\cos \theta} \). This simplifies the problem into two separate limits.

Step 5: Evaluate the First Limit

The first part \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \) is a known result from trigonometric limits.

Step 6: Evaluate the Second Limit

The second part \( \lim_{\theta \to 0} \frac{1}{\cos \theta} \) evaluates to 1 because \( \cos(0) = 1 \), hence \( \frac{1}{\cos \theta} \) approaches \( \frac{1}{1} = 1 \) as \( \theta \) approaches 0.

Step 7: Multiply the Results

Multiply the two results from Step 5 and Step 6: \( 1 \times 1 = 1 \). Therefore, the original limit \( \lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta} \) equals 1.

Key Concepts

Trigonometric Limits

In calculus, trigonometric limits play a crucial role in understanding the behavior of trigonometric functions as they approach certain points. A classic example is the limit of \( \frac{\sin \theta}{\theta} \) as \( \theta \) approaches 0. This limit is pivotal, as it is equal to 1. It sets the foundation for many other limit problems involving trigonometric functions.

This limit is derived using the Squeeze Theorem. It's essential because it simplifies complex trigonometric expressions, making them more manageable. Whenever you encounter a trigonometric limit, it’s beneficial to recall special limits like these. They serve as shortcuts, saving you from evaluating difficult expressions directly.

Additionally, remember that as the angle \( \theta \) gets closer to zero, the behavior of the trigonometric functions can often be predicted or approximated using the aforementioned identities. Embracing these fundamental limits can help unravel otherwise complex calculus problems.

Tangent Function

The tangent function, denoted by \( \tan \theta \), is one of the basic trigonometric functions. It represents the ratio of the opposite side to the adjacent side in a right triangle. Importantly, it can also be expressed using sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This identity is crucial for simplifying expressions involving tangent.

This expression is particularly useful when dealing with limits. For example, when finding \( \lim_{\theta \to 0} \frac{\tan \theta}{\theta} \), substituting \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) transforms the limit into an expression involving \( \sin \theta \) and \( \cos \theta \), which are easier to handle. Such transformations often reveal clearer insights or quicker paths to solutions.

• Using this identity is key when breaking down complex limits into simpler parts.
• It allows you to utilize known limits of sine and cosine to solve problems involving tangent.
• Recognizing tangent in terms of sine and cosine is a common technique in calculus to simplify rational expressions at approach points.

Sine and Cosine Identity

The sine and cosine functions are fundamental trigonometric functions, deeply interlinked by several identities. A particularly useful identity in the context of limits is expressed through tangent: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This identity is pivotal in calculus, as it allows splitting complicated limits into simpler forms.

When you are asked to determine limits involving tangent functions, expressing them in terms of sine and cosine facilities computation. The identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) plays a central role in dividing an expression into parts that can be evaluated separately.

For instance, to solve the problem \( \lim_{\theta \to 0} \frac{\tan \theta}{\theta} \), you reframe it as \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta \cos \theta} \). This separation unveils that it can be tackled by evaluating \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} \) and \( \lim_{\theta \to 0} \frac{1}{\cos \theta} \) individually. These identities make complex calculus problems more approachable and are indispensable tools for simplifying expressions that initially seem daunting.