Problem 11 - 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 \( \lim_{x \to 0} \frac{1 - \cos 2x}{x} = 0.\)

Step 1: Understand the Limit

We need to evaluate the limit \( \lim_{x \to 0} \frac{1 - \cos 2x}{x} \). This involves simplifying the expression so that it can be evaluated at \( x = 0 \).

Step 2: Use Trigonometric Identity

Use the trigonometric identity for \(1 - \cos \theta\): \[ 1 - \cos 2x = 2 \sin^2 x \].Thus, the expression becomes:\[ \lim_{x \to 0} \frac{2 \sin^2 x}{x}. \]

Step 3: Simplify the Expression

Rewrite \( 2 \sin^2 x \) using \( 2 \sin x \cdot \sin x \) and split the limit:\[ \lim_{x \to 0} \frac{2 \sin x}{1} \cdot \lim_{x \to 0} \frac{\sin x}{x}. \]

Step 4: Evaluate Each Limit

We know that \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). Therefore,\[ \lim_{x \to 0} \frac{2 \sin x}{1} \cdot \lim_{x \to 0} \frac{\sin x}{x} = 2 \cdot 1 \cdot 1 = 0. \]

Step 5: Conclusion

The limit evaluates to 0. Thus,\[ \lim_{x \to 0} \frac{1 - \cos 2x}{x} = 0. \]

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that express relationships between trigonometric functions. These identities are essential in simplifying complex expressions, especially in calculus when solving limits. For the exercise given, the identity that connects the expression \( 1 - \cos \theta \) is particularly useful.

The identity \( 1 - \cos 2x = 2 \sin^2 x \) transforms the given limit into a form that's easier to handle. By replacing \(1 - \cos 2x\) with \(2 \sin^2 x\), you simplify the expression significantly. This step leverages the double-angle identity for cosine, which states:

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

Recognizing such identities is crucial for simplifying limits and their evaluation in calculus problems.

Limit Evaluation

Evaluating limits is a core part of calculus and involves finding the value that a function approaches as the input approaches a given point. When faced with evaluating \( \lim_{x \to 0} \frac{1 - \cos 2x}{x} \), you transform the expression using trigonometric identities and algebraic manipulation.

After substituting the identity \( 1 - \cos 2x = 2 \sin^2 x \), you have: \\[ \lim_{x \to 0} \frac{2 \sin^2 x}{x} \] \This fraction can be rewritten by factoring out constants and separating the limit into more manageable parts. This is achieved using the known limit:

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

By splitting the expression into \( \frac{2 \sin x}{1} \times \frac{\sin x}{x} \), observe that the product simplifies efficiently using this result. Understanding the limit evaluation through these steps reduces complexity and highlights the usefulness of foundational limit rules.

Calculus Problem Solving

Solving calculus problems often involves a structured approach to breaking down complex expressions. An effective strategy for tackling the given problem is a step-by-step process:

• Identify mathematical identities and rules applicable to the problem, such as trigonometric identities or basic limits.
• Algebraically manipulate the expression to a simpler, more evaluable form.
• Utilize known limits, such as \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \), for smoother calculation.

Step-by-step problem solving:

• Step 1: Simplifies the original limit using identities.
• Step 2: Breaks it down into sub-expressions with separate limits.
• Step 3: Evaluates these limits using known results or additional calculations.

By following this structured approach, not only do you solve the problem more easily, but you also gain a deeper understanding of fundamental calculus concepts.