Problem 39 - Thomas Calculus

Question

Find the limits in Exercises \(21-42\) $$\lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin 8 x}$$

Step-by-Step Solution

Answer

The limit is \(\frac{3}{8}\).

Step 1: Recognize Trigonometric Limits

Identify the standard trigonometric limits that can be used to simplify the expression. Recognize that as \(x\) approaches 0, both \(\tan 3x\) and \(\sin 8x\) approach 0.

Step 2: Apply Trigonometric Limit Identities

Recall the standard limit identities: \(\lim_{x \to 0} \frac{\tan x}{x} = 1\) and \(\lim_{x \to 0} \frac{\sin x}{x} = 1\). These will help in transforming the given limit.

Step 3: Rewrite the Expression

Rewrite the limit \(\lim_{x \to 0} \frac{\tan 3x}{\sin 8x}\) by multiplying and dividing by \(3x\) in the numerator and \(8x\) in the denominator to create forms matching the standard limits:\[\lim_{x \to 0} \frac{3x}{8x} \cdot \frac{\tan 3x}{3x} \cdot \frac{8x}{\sin 8x}\]

Step 4: Analyze the Components

Analyze the components separately:- \(\lim_{x \to 0} \frac{\tan 3x}{3x} = 1\) by the identity \(\lim_{u \to 0} \frac{\tan u}{u} = 1\).- \(\lim_{x \to 0} \frac{8x}{\sin 8x} = 1\) by the identity \(\lim_{u \to 0} \frac{u}{\sin u} = 1\).Also, \(\lim_{x \to 0} \frac{3x}{8x} = \frac{3}{8}\), since constants can be factored out of the limit.

Step 5: Combine the Results

Combine the results from the analysis of components:\[\lim_{x \to 0} \frac{3}{8} \cdot 1 \cdot 1 = \frac{3}{8}\]Therefore, the limit evaluates to \(\frac{3}{8}\).

Key Concepts

Trigonometric Limits

Understanding trigonometric limits is essential for solving problems that involve functions like sine, cosine, and tangent as the variable approaches certain values, often zero. When dealing with limits of trigonometric functions, one must recognize how these functions behave near critical points. For instance, as \(x\) approaches 0, the values of \(\tan x\) and \(\sin x\) both approach zero, which indicates an indeterminate form of \(\frac{0}{0}\). Knowing the behavior of these trigonometric functions is crucial for simplifying such expressions.
Trigonometric limits are vital in calculus because they help determine the continuity and differentiability of trigonometric functions. A common approach is to transform these indeterminate forms using standard limit identities, which streamline calculations significantly. This leads us to understand why recognizing and using these limits correctly can save time and reduce errors in complex calculus problems.

Limit Identities

Limit identities are fundamental tools used in calculus to simplify the process of evaluating limits, especially for trigonometric functions. There are commonly used standard identities, such as:

• \(\lim_{x \to 0} \frac{\sin x}{x} = 1\)
• \(\lim_{x \to 0} \frac{\tan x}{x} = 1\)

These identities are derived from the limit properties of the sine and tangent functions, which approximate linearity close to zero.
Applying these identities effectively allows us to transform more complex expressions into manageable forms. In the given exercise, knowing these identities helps rewrite the limit expression \(\lim_{x \to 0} \frac{\tan 3x}{\sin 8x}\) by introducing constants into an expression that closely matches the standard forms. This strategic manipulation enables us to substitute the identity values and obtain a numerical result quickly.

Step-by-Step Limit Evaluation

The step-by-step evaluation of limits involves a systematic approach to breaking down the mathematical expression and applying known techniques or identities to solve it. In our exercise, we analyzed an expression where both the numerator and denominator approach zero, creating an indeterminate form.

The process begins by identifying applicable trigonometric limit identities, which help transform the expression into a simpler form by division and multiplication with strategic constants. For example, the expression \(\lim_{x \to 0} \frac{\tan 3x}{\sin 8x}\) is rewritten as \(\lim_{x \to 0} \frac{3x}{8x} \cdot \frac{\tan 3x}{3x} \cdot \frac{8x}{\sin 8x}\).
In the next step, individual limits for the transformed components are evaluated separately:

• \(\lim_{x \to 0} \frac{\tan 3x}{3x} = 1\)
• \(\lim_{x \to 0} \frac{8x}{\sin 8x} = 1\)
• \(\lim_{x \to 0} \frac{3x}{8x} = \frac{3}{8}\)

Finally, these results are combined, yielding the overall limit value of \(\frac{3}{8}\). Following such systematic steps ensures accuracy and a deepened understanding of how the behavior of functions around certain points informs their limits.