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 \( \frac{4}{\sqrt{3}} \).

Step 1: Understand the Function

The function given is \( 2 \sec \frac{x}{3} \). The secant function, \( \sec \theta \), is the reciprocal of the cosine function, \( \frac{1}{\cos \theta} \). Hence, our function becomes \( 2 \cdot \frac{1}{\cos \left(\frac{x}{3}\right)} \).

Step 2: Determine the Point of Interest

We are asked to find the limit as \( x \) approaches \( \frac{\pi}{2} \). This means we're interested in the behavior of the function \( 2 \sec \frac{x}{3} \) as \( x \) gets very close to \( \frac{\pi}{2} \).

Step 3: Substituting the Point into the Function

To gain insight, compute \( \frac{x}{3} \) when \( x = \frac{\pi}{2} \), which gives \( \frac{\pi}{2 \times 3} = \frac{\pi}{6} \). We now compute \( \cos \frac{\pi}{6} \), which equals \( \frac{\sqrt{3}}{2} \). Hence \( \sec \frac{\pi}{6} = \frac{2}{\sqrt{3}} \), and therefore \( 2 \sec \frac{\pi}{6} = 2 \times \frac{2}{\sqrt{3}} = \frac{4}{\sqrt{3}} \).

Step 4: Conclude the Limit

As \( x \) approaches \( \frac{\pi}{2} \), \( \frac{x}{3} \) approaches \( \frac{\pi}{6} \). Since the cosine function is continuous and non-zero at \( \frac{\pi}{6} \), \( \cos \left(\frac{x}{3}\right) \rightarrow \frac{\sqrt{3}}{2} \). Thus, the limit is \( \frac{4}{\sqrt{3}} \).

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is a trigonometric function that is fundamentally related to the cosine function. It is defined as the reciprocal of the cosine function, which means:\[\sec \theta = \frac{1}{\cos \theta}\]Since the cosine function is associated with the horizontal coordinate of a point on the unit circle, the secant function essentially measures how _"wide"_ or _"tall"_ that will need to be for the secant function to maintain the reciprocal property.
For example:

• If \( \cos \theta = \frac{1}{2} \), then \( \sec \theta = 2 \).
• If \( \cos \theta \) approaches zero, \( \sec \theta \) will tend towards infinity—or become undefined, indicating the vertical asymptotes in the graph of the secant function.

This property of approaching infinity illustrates why the secant function has distinctive vertical asymptotes and fluctuates towards large positive or negative values where the cosine function is equal to zero. Understanding this aspect can aid in determining limits that involve secant.

Trigonometric Limits

Trigonometric limits involve understanding how trigonometric functions behave as their input values tend towards certain numbers, often resulting in the function approaching an asymptote or a specific finite value.
In this example, we are exploring the limit:\[\lim _{x \rightarrow \pi / 2} 2 \sec \frac{x}{3}\]We consider \( \sec \frac{x}{3} \) as \( x \) approaches \( \pi/2 \). By substituting \( x = \pi/2 \) into \( \frac{x}{3} \), we find:\[\frac{\pi}{2 \times 3} = \frac{\pi}{6}\]Knowing that:

• \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)
• \( \sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}} = \frac{2}{\sqrt{3}} \)

it follows that the limit evaluates to:\[2 \sec \frac{\pi}{6} = \frac{4}{\sqrt{3}}\] Such problems often hinge on the reciprocal property of secant and the simplicity provided by special angles like \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \) in unit circle trigonometry.

Continuous Functions

A function is continuous if, roughly speaking, you can draw its graph without lifting your pen off the paper. Mathematically, a function \( f(x) \) is continuous at a point \( c \) if three conditions are satisfied:

• \( f(c) \) is defined.
• The limit of \( f(x) \) as \( x \rightarrow c \) exists.
• The limit of \( f(x) \) as \( x \rightarrow c \) equals \( f(c) \).

This concept supports our understanding of finding limits in functions involving secant, like in our exercise. As \( x \) nears \( \pi/2 \), \( \sec \frac{x}{3} \) breaks down into \( \frac{1}{\cos \left(\frac{x}{3}\right)} \). Since cosine is continuous for real numbers, particularly at \( \frac{\pi}{6} \), and does not equate to zero, the secant function maintains continuity within the context.
Why is this important? When tackling problems that involve limits approaching angle values, the continuity of the function assists us in confirming that limits can be directly evaluated using substitution when within the domain that secant and cosine tolerate. This results in precise computations without worrying about undefined behavior for simple substitutions.