Problem 10 - Calculus for Biology and Medicine

Question

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x^{2}} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify Trigonometric Identity

To evaluate the limit, we first apply a well-known trigonometric identity to simplify the numerator. Recognize that \(1 - \cos^2 x\) is equivalent to \(\sin^2 x\) based on the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\).

Step 2: Rewrite the Limit with Simplified Expression

Using the identity, rewrite the original limit as \(\lim _{x \rightarrow 0} \frac{\sin^2 x}{x^2}\).

Step 3: Apply Standard Limit Property

Use the standard limit property \(\lim_{x\to 0} \frac{\sin x}{x} = 1\). Rewrite the expression as \(\lim _{x \rightarrow 0} \left(\frac{\sin x}{x}\right)^2\).

Step 4: Evaluate the Limit

Based on the standard limit property, since \(\lim_{x\to 0} \frac{\sin x}{x} = 1\), it follows that \(\lim_{x\to 0} \left(\frac{\sin x}{x}\right)^2 = 1^2\). Therefore, the limit is \(1\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is central to understanding many trigonometric expressions. In trigonometry, one of the most fundamental identities is given by \( \sin^2 x + \cos^2 x = 1 \). This formula expresses a fundamental truth about the relationships between sine and cosine functions for any angle \( x \).

In the context of our exercise, we use this identity to transform the numerator. The expression \( 1 - \cos^2 x \) can be rewritten as \( \sin^2 x \). By substituting this into our limit, it becomes easier to manage and simplify.

Without this identity, converting complex expressions into solvable limits would be significantly more challenging. It emphasizes how foundational trigonometric identities like the Pythagorean Identity are in simplifying calculus problems.

Standard Limit Property

When dealing with limits, especially those involving trigonometric functions, certain standard limit properties are invaluable. A particularly important one is \( \lim_{x\to 0} \frac{\sin x}{x} = 1 \). This property gives us a powerful tool for evaluating limits involving sine functions.

In our trigonometric limit problem, substituting \( 1 - \cos^2 x \) with \( \sin^2 x \) allows us to rewrite the limit as \( \lim _{x \rightarrow 0} \frac{\sin^2 x}{x^2} \). This is the same as saying \( \left(\frac{\sin x}{x}\right)^2 \). Since \( \frac{\sin x}{x} \to 1 \) as \( x \to 0 \), squaring it gives \( 1^2 = 1 \).

Recognizing and applying this standard limit property streamlines the process of solving limits, reducing complex expressions to simpler forms. It's a time-tested technique taught early in calculus courses to facilitate dealing with a wide range of problems.

Trigonometric Identities

Trigonometric identities are the building blocks of trigonometry. They provide essential shortcuts and simplifications for a wide variety of equations involving angles. In our example, beyond the Pythagorean Identity, we rely on the knowledge that these identities can transform challenging problems into solvable ones.

Trigonometric identities are not just limited to sine and cosine. They include a variety of functions and their relationships, such as tangent, secant, and cotangent identities. These relationships provide essential tools for verifying and solving equations by allowing substitution and simplification.

In calculus, particularly when evaluating limits, knowing your trigonometric identities can make solving problems much more feasible. They offer a map through the dense forest of complex trigonometric expressions, leading toward solutions that might not be immediately obvious without them.