Problem 17 - Calculus for Biology and Medicine

Question

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos (5 x)}{2 x} $$

Step-by-Step Solution

Answer

\( \frac{25}{4} \)

Step 1: Identify the Problem Type

We are tasked with evaluating a trigonometric limit as \( x \) approaches 0. The expression includes trigonometric functions, specifically the cosine function, which suggests using trig identities might be helpful.

Step 2: Apply Trigonometric Identity

Recognize that the identity \( 1 - \cos(\theta) = 2\sin^2(\theta/2) \) can be applied here. Rewrite \( 1 - \cos(5x) \) using this identity, so that \( 1 - \cos(5x) = 2\sin^2(5x/2) \). Substitute this into the limit expression.

Step 3: Rewrite the Limit Expression

Substitute the identity into the limit: \[ \lim _{x \rightarrow 0} \frac{1-\cos (5 x)}{2 x} = \lim _{x \rightarrow 0} \frac{2\sin^2(5x/2)}{2 x} = \lim _{x \rightarrow 0} \frac{\sin^2(5x/2)}{x} \]

Step 4: Manipulate and Separate Components

Rewrite the expression further: \[ \lim _{x \rightarrow 0} \frac{\sin^2(5x/2)}{x} = \lim _{x \rightarrow 0} \left( \frac{\sin(5x/2)}{x} \cdot \sin(5x/2) \right) \] Now separate the limit into the product of two limits for easier evaluation.

Step 5: Evaluate the Separate Limits

Using the known limit \( \lim_{u \rightarrow 0} \frac{\sin(u)}{u} = 1 \), set \( u = 5x/2 \), hence this also implies \( \lim_{x \rightarrow 0} \frac{\sin(5x/2)}{5x/2} = 1 \). Apply this result to compute:\[ \lim_{x \rightarrow 0} \left( \frac{\sin(5x/2)}{x} \right) = \lim_{x \rightarrow 0} \left( \frac{\sin(5x/2)}{5x/2} \cdot \frac{5}{2} \right) = 1 \cdot \frac{5}{2} = \frac{5}{2} \]

Step 6: Compute the Final Limit

Since \( \lim_{x \rightarrow 0} \frac{\sin(5x/2)}{x} = \frac{5}{2} \), the limit of the square, \( \lim_{x \rightarrow 0} \sin^2(5x/2)/x \), is:\[ \left( \frac{5}{2} \right)^2 = \frac{25}{4} \]Hence, \( \lim_{x \rightarrow 0} \frac{1-\cos (5 x)}{2 x} = \frac{25}{4} \).

Key Concepts

Cosine Function

The cosine function, often written as \( \cos(\theta) \), is a fundamental part of trigonometry. It represents the horizontal coordinate of a point on the unit circle corresponding to an angle \( \theta \). The cosine function has a range from -1 to 1. When you see an expression like \( \cos(5x) \), it means you're dealing with the cosine of five times the angle \( x \).
Understanding this function helps us in dealing with many trigonometric problems, especially when combined with trigonometric identities. One of the identities we often use is:

• \( 1 - \cos(\theta) = 2\sin^2(\theta/2) \)

This identity allows us to express the difference from 1 as a squared sine term.
Applying these identities appropriately simplifies complex expressions and is vital in evaluating limits involving trigonometric functions.

Limit Evaluation

Limit evaluation is a crucial concept in calculus. Limits help us understand the behavior of functions as the input approaches a particular value—in this case, as \( x \) approaches 0.
For trigonometric limits, particularly those involving functions like sine and cosine, certain known limits make the evaluation straightforward. One such identity is:

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

This identity simplifies trigonometric limits where sine terms are involved. For the limit problem we're solving, \( \lim_{x \to 0} \frac{1-\cos(5x)}{2x} \), we use this identity to transform the trigonometric function expression into a more manageable form.
When evaluating such a limit, recognizing patterns and using trigonometric identities play a significant role in simplifying expressions and achieving the solution.

Trigonometric Identity

Trigonometric identities are equations that relate different trigonometric functions and are true for all values of the involved variables. They allow us to rewrite trigonometric expressions in different, often more simplified forms.

• For example, \( 1 - \cos(\theta) = 2\sin^2(\theta/2) \) is a pivotal identity for converting a cosine expression into a sine one.
• This identity enables us to transform the complicated expression \( 1 - \cos(5x) \) into something easier to manage: \( 2\sin^2(5x/2) \).

By substituting these identities into our equations, we simplify the limit calculation process.
Therefore, having a strong grasp of trigonometric identities can dramatically ease solving limits and other calculus problems involving trigonometric functions. Identifying when and how to use these identities is a skill developed with practice and understanding.