Problem 16 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Recognize the Trigonometric Identity

The given limit \( \lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x} \) can be simplified using the Pythagorean identity for cosine: \( \sin^2 x = 1 - \cos^2 x \). Therefore, we can rewrite the expression as \( \frac{\sin^2 x}{x} \).

Step 2: Apply Standard Limit

Recall the standard limit \( \lim _{x \rightarrow 0} \frac{\sin x}{x} = 1 \). Although we have \( \sin^2 x \), we need the expression to match the standard limit format. Consider rewriting \( \sin^2 x \) as \( (\sin x) \cdot (\sin x) \).

Step 3: Factor and Simplify the Limit Expression

Split \( \frac{\sin^2 x}{x} \) into two separate fractions: \( \sin x \cdot \frac{\sin x}{x} \). This allows us to apply the standard limit separately.

Step 4: Evaluate the Split Limit

Use the fact that \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). Thus, the limit \( \lim_{x \to 0} [\sin x \cdot \frac{\sin x}{x}] = \lim_{x \to 0} \sin x \cdot \lim_{x \to 0} \frac{\sin x}{x} = 0 \cdot 1 = 0 \).

Step 5: Conclusion

Thus, the original limit \( \lim_{x \to 0} \frac{1-\cos^2 x}{x} = 0 \).

Key Concepts

Pythagorean Identity in Trigonometry

The Pythagorean identity is a fundamental concept in trigonometry. It relates the sine and cosine functions, giving us a way to express one in terms of the other. This identity states:

• \( \sin^2 x + \cos^2 x = 1 \)

This relationship is extremely useful for simplifying trigonometric expressions, particularly when evaluating limits. In the context of the original exercise, we see it allow us to transform \( 1 - \cos^2 x \) into \( \sin^2 x \). This simplification is crucial because it highlights how the expression can be maneuvered towards a more recognizable form for limit evaluation. Remembering this identity can make tackling trigonometric limits much simpler. With practice, recognizing when to use it will become second nature.

Standard Limit Involving Sine

In calculus, there is a well-known standard limit involving the sine function that is extremely handy when dealing with problems that involve trigonometric functions approaching zero. This standard limit is given by:

• \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \)

This formula tells us that as \( x \) approaches zero, the ratio of the sine of \( x \) to \( x \) approaches one. It’s important to understand that this result comes from the properties of the sine function and its behavior near zero. In the exercise, since we are dealing with \( \sin^2 x \), we break it down into \( \sin x \cdot \sin x \) to make use of the standard limit. Recognizing these basic forms can greatly simplify limit evaluation.

Limit Evaluation Technique

Evaluating limits, especially trigonometric ones, often involves a series of clever manipulations to bring an expression into a more familiar form. For example, in the given exercise, after using the Pythagorean identity, we needed to split the expression \( \frac{\sin^2 x}{x} \) into two parts:

• \( \sin x \) and \( \frac{\sin x}{x} \)

This allowed us to evaluate each part's limit independently. The concept of product limits applies here; if the limit of a product exists, you can evaluate it as the product of the limits. So, we calculate:

• \( \lim_{x \to 0} \sin x = 0 \)
• \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \)

This step-by-step breakdown allows us to conclude that the original limit equals zero. By understanding these techniques and fundamentals, you become adept at breaking complex expressions into simpler parts, making the evaluation process more intuitive.