Question

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 7} \sec \left(\frac{\pi x}{6}\right) $$

Short answer

The limit of the trigonometric function as \(x\) approaches 7 is \(-2\sqrt{3}/3\).

Step-by-step solution

Understand the Problem

We are asked to find the limit as \(x\) approaches 7 for the function \(\sec \left(\frac{\pi x}{6}\right)\). The function is a trigonometric function, specifically the secant function. We need to substitute \(x = 7\) into the function and find the value of the function at that point.

Substitute the value into the function

Let's substitute \(x = 7\) into the function. This gives us: \(\sec \left(\frac{\pi \cdot 7}{6}\right)\).

Evaluate the function

Now we need to evaluate this function. We have: \(\sec \left(\frac{7\pi}{6}\right)\). We know that \(\sec(x) = \frac{1}{\cos(x)}\), so we can rewrite the function as: \(\frac{1}{\cos \left(\frac{7\pi}{6}\right)}\). Evaluating the cosine term inside the function gives us: \(-\sqrt{3}/2\). So the final value of the function is: \(\frac{1}{-\sqrt{3}/2} = -2/\sqrt{3}\).

Simplification

To make our answer look cleaner, let's rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{3}\). This yields: \(-2\sqrt{3}/3\).

Key concepts

Secant Function

The secant function, denoted as \( \sec(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function. This means that \( \sec(x) = \frac{1}{\cos(x)} \). Understanding this relationship is crucial because it highlights how the secant function is directly tied to the cosine function. Unlike cosine, which ranges between -1 and 1, secant can take on values less than -1 or greater than 1. This arises because as cosine approaches zero, the secant value becomes very large — hinting at its vertical asymptotes. In practice, knowing how to express secant in terms of cosine is vital when evaluating trigonometric limits, especially when dealing with specific angles or intervals.

Trigonometric Limits

Trigonometric limits might seem intimidating at first, but they guide us in understanding how trigonometric functions behave as inputs approach certain values. The key to solving these problems is understanding the basic properties of trigonometric functions like sine, cosine, and their reciprocals — secant, cosecant, tangent, and cotangent.

• When evaluating a limit of a trigonometric function, often, we substitute the point directly into the function to determine the limit.
• If direct substitution results in an undefined value, we might need to manipulate the function using trigonometric identities or other mathematical tools.

In this specific problem, we found the limit of \( \sec \left(\frac{\pi x}{6}\right) \) as \( x \) approaches 7. Substituting yielded \( \sec \left(\frac{7\pi}{6}\right) \), which we simplified using the secant-cosine relationship. Understanding and applying these techniques are invaluable for solving similar limits with trigonometric functions.

Cosine Function

The cosine function is one of the primary trigonometric functions, often written as \( \cos(x) \). It provides the ratio of the adjacent side to the hypotenuse in a right-angled triangle. This function is periodic with a period of \( 2\pi \), which means it repeats its values every \( 2\pi \) units along the x-axis.

• Its range is between -1 and 1, making it a bounded function, unlike secant which has larger amplitudes due to its reciprocal nature.
• Cosine is an even function, so \( \cos(-x) = \cos(x) \), reflecting its symmetric nature about the y-axis.

In our limit problem, the value of \( \cos \left(\frac{7\pi}{6}\right) \) was found to be \(-\frac{\sqrt{3}}{2}\). Recognizing these specific angle values and their corresponding cosine values is essential, especially for evaluating expressions that involve reciprocal trigonometric functions like secant. Remembering key angles and their cosine values can significantly ease the process of solving trigonometric limits.