Question

Find the limit. $$ \lim _{x \rightarrow 5 \pi / 3} \cos x $$

Short answer

The limit of the function as x approaches \(5\pi / 3\) is 1/2.

Step-by-step solution

Understand the problem

The problem is to find a value of the limit of the function as x approaches \(5\pi/3\). This means we are looking for the value that the function approaches as x gets closer and closer to \(5\pi/3\). In this case, the function given is \(\cos x\).

Substituting the value in the function

Because this is a straight-forward limit, we just substitute the approaching value (\(5\pi/3\)) into the cosine function. Therefore the solution of the problem is \(\cos(5\pi/3)\).

Calculate the cosine value

The value of \(\cos(5\pi/3)\) equals 1/2.

Key concepts

Cosine Function

The cosine function, denoted as \(\cos x\), is a fundamental concept in trigonometry, which itself is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

As a trigonometric function, the cosine also applies to the unit circle, where it represents the x-coordinate of a point on the circle corresponding to a given angle. It is a periodic function, meaning it repeats its values in a consistent pattern at regular intervals, specifically every \(2\pi\) radians, or 360 degrees. When dealing with the limits involving a cosine function, it's important to remember its periodic nature and the fact that it always lies between -1 and 1 inclusive.

Here's a visualization of the cosine function's behavior:

• \(\cos 0 = 1\)
• \(\cos \frac{\pi}{2} = 0\)
• \(\cos \pi = -1\)
• \(\cos \frac{3\pi}{2} = 0\)
• \(\cos 2\pi = 1\)

This periodicity is key when calculating the limits for angles that may extend beyond the basic first period of \(0\) to \(2\pi\).

Calculus

Calculus is a branch of mathematics that studies how things change. Its two main branches, differential calculus and integral calculus, focus on the rate of change (slopes and curves) and accumulation, respectively. In the context of this exercise, we are tackling a concept that falls under differential calculus: the limit.

In simple terms, a limit tries to find out what value a function approaches as the input (or 'x') approaches a certain value. This does not always mean that the function actually reaches this value, but rather what value it would approximate if it could continue. For example, when you approach a traffic light, you would slow down (decreasing speed is the function) as you approach the light (the point at which 'x' equals the traffic light position).

When we calculate a limit, we're really asking, 'What value does the function get closer and closer to, as 'x' gets infinitely close to a certain number?'. It is a fundamental concept that underpins many areas of calculus, including continuous functions, derivatives, and integrals. It deals with the behavior of functions and is crucial for understanding and solving practical problems in science and engineering.

Trigonometric Limits

Trigonometric limits are a specific kind of limit problem found in calculus that involve trigonometric functions such as sine, cosine, and tangent. When calculating the limit of these functions as 'x' approaches a certain value, you must take into account their oscillating nature and periodicity.

For instance, in the task of finding \(\lim _{x \rightarrow 5 \pi / 3} \cos x\), we are looking at the behavior of the cosine function as the angle 'x' approaches \(5\pi/3\) radians. Because the cosine function is periodic, we know that the value it approaches must be one of the values that cosine can take, somewhere between -1 and 1.

In many cases, there is no need for complex calculations because the value of the limit can be directly determined by the known values of trigonometric functions at specific angles. This is the case for angles that correspond to the standard positions on the unit circle, where the cosine values are well established. By recognizing these patterns and standard values, solving trigonometric limits often becomes a process of substitution and simplification.