Question

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 1} \cos \frac{\pi x}{3} $$

Short answer

The limit of the function \(\cos \frac{\pi x}{3}\) as \(x\) approaches \(1\) is \(\frac{1}{2}\)

Step-by-step solution

Understand the function limits

Firstly, understand that the limit \(\lim _{x \rightarrow a} f(x)\) of a function at a point can be evaluated by substituting the point into the function if the function is continuous at that point. In this case, \(\cos \frac{\pi x}{3}\) is continuous at every point.

Substitute the limit point into the function

Now, substituting \(x = 1\) into the function, one obtains \(\cos \frac{\pi (1)}{3}\) = \(\cos \frac{\pi}{3}\)

Evaluate the cosine function

Finally, to solve this expression, use the fact that \(\cos \frac{\pi}{3}\) equals \( \frac{1}{2}\) in terms of cosine values. This is a fact one should know from trigonometry.

Key concepts

Evaluating Limits

When it comes to evaluating limits of functions, the main goal is to determine what value a function approaches as the input (usually denoted as 'x') gets closer to a certain point. The notation for expressing this concept is \( \lim _{x \rightarrow a} f(x) \), which reads as 'the limit of f(x) as x approaches a'. To evaluate a limit, several methods can be used, including direct substitution, factoring, rationalization, and employing L'Hôpital's Rule, among others.

In cases where the function is continuous at the point in question, determining the limit is as straightforward as substituting that point into the function. The limit and the function value are the same at points of continuity. For our exercise, as the cosine function is continuous everywhere, the limit as x approaches 1 is simply found by plugging in x=1 into the cosine function.

Continuity of Functions

Understanding the continuity of functions is essential in the evaluation of limits. A function is said to be continuous at a point if the following three conditions are met: (1) the function is defined at the point, (2) the limit of the function exists at the point, and (3) the limit of the function as x approaches the point is equal to the function's value at the point.

Continuity ensures that there are no abrupt changes or 'jumps' in the function's behavior around the point, making direct substitution an effective tool for finding limits. For example, trigonometric functions such as sine and cosine are continuous over their entire domains, enabling us to directly substitute values to evaluate limits.

Trigonometric Limits

When dealing with trigonometric limits, we often encounter functions involving sine, cosine, tangent, and other trigonometric ratios. To evaluate trigonometric limits, one should be familiar with the basic limits, such as \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \) and the periodicity and symmetries of trigonometric functions.

For instance, knowing that the cosine function has a period of \( 2\pi \) allows us to predict its behavior as the input variable approaches any multiple of \( 2\pi \) and thus helps to simplify the process of limit evaluation. Moreover, the commonly known values at specific angles (like \( \frac{\pi}{3}, \frac{\pi}{4}, \frac{\pi}{6} \)) are applicable shortcuts to calculating such limits.

Cosine Values in Trigonometry

In the realm of cosine values in trigonometry, it's important to recognize that the cosine function varies between -1 and 1 and has specific values for certain standard angles.

These angles, frequently in radians, include \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \( \pi \), and are key to quickly evaluating trigonometric functions without calculators. For example, \( \cos \frac{\pi}{6} = \sqrt{3}/2 \) and as seen in the given exercise, \( \cos \frac{\pi}{3} = 1/2 \). Memorizing these values and understanding the unit circle where these cosine values are graphically represented, can greatly facilitate the process of evaluating trigonometric limits.