Question

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 0} \sec 2 x $$

Short answer

The limit of \( \sec 2x \) as \( x \) approaches 0 is 1.

Step-by-step solution

Understanding the secant function

The secant function, \( \sec x \), is equivalent to \( \frac{1}{\cos x} \). This means \( \sec 2x = \frac{1}{\cos 2x} \).

Substituting the limit

Substitute \( x = 0 \) into the modified function \( \frac{1}{\cos 2x} \). This gives \( \frac{1}{\cos 2(0)} = \frac{1}{\cos 0} \).

Evaluating the cosine function

\( \cos 0 \) is 1, so \( \frac{1}{\cos 0} \) is \( \frac{1}{1} \), which is 1.

Key concepts

Secant Function

The secant function, denoted as \( \text{sec}(x) \), may not be as familiar to students as its cosine counterpart, but it plays a significant role in trigonometry. Defined as the reciprocal of the cosine function, \( \text{sec}(x) = \frac{1}{\text{cos}(x)} \), it implies that whenever you are asked to find the value of a secant function, you are essentially tasked with finding the reciprocal of the corresponding cosine value.

For example, \( \text{sec}(2x) \) means you're looking at the reciprocal of \( \text{cos}(2x) \). In the context of the given problem where the limit as \( x \) approaches zero is sought, understanding this reciprocal relationship is crucial. The behavior of the secant function near \( x = 0 \) can be directly deduced from the behavior of the cosine function at the same point, given that they are inversely related.

Evaluating Cosine

The cosine function is one of the basic trigonometric functions and depicts the horizontal coordinate of a point on the unit circle. When you evaluate the cosine function at different values, you're essentially finding out the x-coordinate of the point on the unit circle that corresponds to the angle made with the positive x-axis. For well-known angles, these values are typically memorized or easily found using a unit circle.

For instance, \( \text{cos}(0) \) corresponds to the angle of 0 radians, where the point on the unit circle is at \( (1, 0) \). Therefore, the cosine value is 1. In our problem, evaluating \( \text{cos}(2 \times 0) \) simplifies to \( \text{cos}(0) \), which as mentioned, is 1. This direct substitution method is often utilized when dealing with limits in trigonometric functions, especially when the limit approaches a common angle.

Trigonometric Limits

Understanding limits within the context of trigonometry involves grasping how trigonometric functions behave as the input values approach a certain number or infinity. Limits can tell us about the behavior of a trigonometric function even when direct substitution isn't possible due to indeterminate forms like 0/0.

In cases where direct substitution is possible, evaluating trigonometric limits becomes much simpler, as seen in our example where the limit of \( \text{sec}(2x) \) as \( x \) approaches 0 is straightforward. However, when functions approach indeterminate forms, other techniques such as L'Hôpital's rule or trigonometric identities may be necessary. In the broader scope of calculus, mastering limits in trigonometry paves the way for understanding how these functions behave in derivatives and integrals, which are fundamental to the study of continuous change.