Problem 64 - Calculus of a Single Variable

Question

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 0} \frac{3(1-\cos x)}{x} $$

Step-by-Step Solution

Answer

The limit of the function as x approaches 0 is 0.

Step 1: Identify Form of Function

The function \(\frac{{3(1-\cos x)}}{{x}}\) resembles the identity \(\lim_{{x \rightarrow 0}} \frac{{1-\cos x}}{{x}} = 0\). If we can rewrite the function in the form of this identity, we will simplify solving the limit.

Step 2: Rewrite the Function

We can rewrite the function by changing the 3 in front of \(1 - \cos x\), giving us \(3 \times \frac{{1-\cos x}}{x}\). Now we have the function in the form of the identity plus a constant 3.

Step 3: Apply Limit Identity

Now we apply the limit identity to get the limit. The limit of \(3 \times \frac{{1-\cos x}}{x}\) as \(x\) approaches 0 is thus \(3 \times \lim_{x \rightarrow 0} \frac{{1-\cos x}}{x}\). Using the limit identity, this equals to \(3 \times 0)\).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with the properties and changes of continuous functions. One of the fundamental operations in calculus is finding the limit of a function as its input approaches a specific value. This process is crucial for understanding the behavior of functions at points where they may not be well-defined, such as discontinuities or points of indeterminacy.

When calculating the limit of a function like \( \lim_{x \rightarrow a} f(x) \) as \(x\) approaches the value \(a\), we are essentially asking what value the function \(f(x)\) gets closer to as \(x\) gets arbitrarily close to \(a\). In practice, this often involves simplifying the expression for \(f(x)\) so that the limit can be evaluated directly or applying known limit properties and identities to find the result.

Limit Identities

Limit identities are specialized results in calculus that describe the limit of certain types of functions as variable approaches a particular point. These identities are exceptionally useful for finding the limits of more complex functions that incorporate these patterns.

One common limit identity is \( \lim_{x \rightarrow 0} \frac{1 - \cos(x)}{x} = 0 \). This identity allows us to evaluate the limit of functions that resemble this form without having to perform detailed algebraic manipulations each time. In the provided exercise, identifying that the given function \( \frac{3(1 - \cos(x))}{x} \) is a multiple of this identity structure empowers us to simplify the process. By factoring out the constant and applying the identity, we can solve the limit quickly and accurately.

Trigonometric Limits

Trigonometric limits refer to the limits of functions that involve trigonometric expressions. Evaluating these limits often requires the use of specific trigonometric identities or limit properties unique to trigonometric functions.

Understanding the behavior of trigonometric functions as their input approaches a particular value is important in various applications of calculus, from graphing function behavior to solving real-world problems involving periodic phenomena. For instance, knowing that the sine function oscillates between \(-1\) and \(1\) can inform us about the potential limits of functions involving \(\sin(x)\) as \(x\) approaches infinity.

The exercise at hand \( \lim _{x \rightarrow 0} \frac{3(1-\cos x)}{x} \) is an example of a trigonometric limit. The limit is not immediately evident because approaching 0 introduces a form of indeterminate nature (0 in the denominator). However, by utilizing the known limit behavior of a related trigonometric identity, we can arrive at the conclusion without requiring the trigonometric function to be evaluated directly at the point where it's undefined.