Problem 7 - Calculus for Biology and Medicine

Question

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow \pi / 2} 2 \sec \frac{x}{3} $$

Step-by-Step Solution

Answer

The limit is \( \frac{4\sqrt{3}}{3} \).

Step 1: Understanding the Secant Function

The secant function is the reciprocal of the cosine function. Thus, \( \sec \theta = \frac{1}{\cos \theta} \). To evaluate the limit \( \lim _{x \rightarrow \pi / 2} 2 \sec \frac{x}{3} \), we first need to understand that \( \sec \frac{x}{3} = \frac{1}{\cos \frac{x}{3}} \).

Step 2: Substitute the Limit Point into the Function

Substitute \( x = \frac{\pi}{2} \) into the angle of the cosine function: \( \frac{x}{3} = \frac{\pi}{6} \). This means our limit expression becomes \( 2 \cdot \frac{1}{\cos \frac{\pi}{6}} \).

Step 3: Evaluate \( \cos \frac{\pi}{6} \)

From trigonometry, we know that \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \). Therefore, \( \sec \frac{\pi}{6} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \).

Step 4: Calculate the Limit Expression

Now substitute \( \sec \frac{\pi}{6} = \frac{2}{\sqrt{3}} \) back into our limit expression: \( 2 \cdot \frac{2}{\sqrt{3}} = \frac{4}{\sqrt{3}} \).

Step 5: Simplify the Result

Simplify \( \frac{4}{\sqrt{3}} \) by multiplying by \( \frac{\sqrt{3}}{\sqrt{3}} \) to rationalize the denominator: \( \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{3} \).

Key Concepts

Trigonometric Limits

Trigonometric limits are a foundational concept in calculus, often encountered when working with limits in functions involving trigonometric operations. As you approach these limits, you typically substitute values close to a specific point into trigonometric expressions. It's crucial to understand how functions like sine, cosine, and tangent behave in the vicinity of particular angles, which may involve using known trigonometric values or identities.

When solving problems like \( \lim_{x \rightarrow \pi / 2} 2 \sec \frac{x}{3} \), the goal is to simplify the expression as much as possible, often using known trigonometric values or properties. Trigonometric identities allow you to break down complex functions into simpler parts, facilitating easier evaluation of limits.

Common scenarios include limits that require using the small-angle approximations or the Pythagorean identity to resolve forms that arise when dealing with angles close to specialty angles like \( \pi / 2 \), \( \pi / 6 \), etc. This exercise provides a fine example of employing trigonometric values at specific points to achieve an exact limit result.

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function. Mathematically, this relationship can be expressed as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This reciprocal relationship is crucial when evaluating limits that involve the secant function, especially when cosine values are known. As shown in the step-by-step solution, the function \( \sec \frac{x}{3} \) needs transforming into \( \frac{1}{\cos \frac{x}{3}} \).

Understanding the secant function's behavior around critical angles is essential. Since \( \sec \theta \) becomes undefined where \( \cos \theta = 0 \), knowing these critical points helps avoid complications during evaluation. Proficiency with secant values for common angles, like \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), strengthens one's ability to independently solve limit problems involving these functions.

Limit Evaluation Steps

Evaluating limits step by step is a methodical approach that simplifies complex mathematical expressions. Here's a breakdown of how to apply these steps using our example problem.

**Step 1: Recognize the Function**
Identifying the underlying trigonometric function is the first step. Here, it's \( \sec \frac{x}{3} \), meaning you'll work with its reciprocal function, cosine.

**Step 2: Substitution of the Limit Point**
Plug the limit point into your trigonometric identity. Replacing \(x = \frac{\pi}{2} \), find \( \frac{x}{3} = \frac{\pi}{6} \). This substitution simplifies the problem significantly by allowing direct calculation of known trigonometric values.

**Step 3: Calculate Trigonometric Values**
Using trigonometric tables or knowledge, determine \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \). Convert this into \( \sec \frac{\pi}{6} \) using the secant definition, resulting in \( \frac{2}{\sqrt{3}} \).

**Step 4: Solve and Simplify**
Finish by plugging back into the original expression and simplifying. Multiply through, if needed, and rationalize denominators for tidy results.

By systematically applying these steps, you eliminate guesswork and build a robust process for solving a wide array of limit problems. This procedure not only clarifies individual components but also ensures you can confidently tackle similar challenges in calculus.