Problem 24 - Thomas Calculus

Question

Find the limits in Exercises \(21-42\) $$\lim _{y \rightarrow 0} \frac{\sin 3 y}{4 y}$$

Step-by-Step Solution

Answer

The limit is \(\frac{1}{4}\).

Step 1: Identify the Indeterminate Form

As we substitute the limit point, that is, as \(y\) approaches 0, the expression takes the form \(\frac{0}{0}\). This is an indeterminate form that suggests further manipulation is necessary to find the limit.

Step 2: Use the Standard Limit Formula

Recall the standard limit formula: \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). This formula will help simplify our expression by finding a similar structure.

Step 3: Rewrite the Expression

In order to use the standard limit formula, adjust the expression \( \lim_{y \to 0} \frac{\sin 3y}{4y} \) so that it matches the structure of \( \frac{\sin x}{x} \). Notice the factor of 3 inside the sine function.

Step 4: Factor and Simplify

Rewrite the expression to incorporate the standard limit:\[\lim_{y \to 0} \frac{\sin 3y}{4y} = \lim_{y \to 0} \frac{3 \sin 3y}{12y} = \frac{1}{4} \cdot \lim_{y \to 0} \frac{\sin 3y}{3y}\]Separate the coefficient \(\frac{3}{12}\), which simplifies to \(\frac{1}{4}\), from the limit.

Step 5: Apply the Standard Limit

Now apply the standard limit to the expression:\[\lim_{y \to 0} \frac{\sin 3y}{3y} = 1\]Then, substitute this back into the factored expression:\[\frac{1}{4} \times 1 = \frac{1}{4}\]

Step 6: Conclusion

Therefore, the limit is:\[\lim_{y \to 0} \frac{\sin 3y}{4y} = \frac{1}{4}\]

Key Concepts

Indeterminate Forms

When evaluating limits, an expression like \( \frac{0}{0} \) is known as an indeterminate form. This occurs when substituting a limit point, such as \( y \rightarrow 0 \), results in both the numerator and denominator equating to zero. This form doesn't provide enough information to determine the limit directly, as it lacks a defined numerical value. Instead, it indicates that further algebraic manipulation or limit evaluation techniques are necessary to resolve the uncertainty and find the actual limit.

Indeterminate forms are essential for identifying situations where typical substitution won't yield a clear answer. They serve as a signal that you need to dive deeper into calculus methods, like algebraic manipulation, factoring, or applying standard limits, to reach a solution. Understanding and recognizing indeterminate forms is a critical step in mastering calculus limits.

Trigonometric Limits

Trigonometric limits often involve expressions containing sine, cosine, or other trigonometric functions. A particularly useful concept in these cases is the standard limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). This foundational limit is a valuable tool for evaluating limits involving sine within these specific kinds of expressions.

When faced with trigonometric limits, the goal is often to rewrite the expression in a way that resembles this standard form. By recognizing patterns and structures in the problem that align with known limits, you can simplify the process considerably. For example, in the expression \( \lim_{y \to 0} \frac{\sin 3y}{4y} \), the presence of \( \sin 3y \) suggests using the standard limit.

• This involves adjusting constants and factors to create the familiar \( \frac{\sin x}{x} \) structure.
• By manipulating these expressions, you transform a seemingly complex limit into something straightforward to evaluate, ensuring efficient problem-solving.

Limit Evaluation Techniques

To evaluate limits effectively, especially those involving indeterminate forms or trigonometric functions, certain techniques are crucial. Here are some techniques that can simplify these problems:

• **Factoring**: Break down complex expressions into simpler parts that can be reduced or canceled.
• **Substitution**: Sometimes, a clever substitution can simplify an expression, making it easier to evaluate the limit.
• **Multiplying by a Conjugate**: In cases involving radicals, multiplying by the conjugate can help eliminate the radical, simplifying the expression.

In our exercise, one technique involved factoring the original expression to match the form of known trigonometric limits. By recognizing that multiplying by \( \frac{3}{3} \) transforms \( \frac{\sin 3y}{4y} \) into a convenient form, simplicity is achieved. This manipulation allows straightforward application of standard limits, streamlining the calculation and leading to the correct solution. Understanding and applying these techniques is fundamental to becoming proficient in limit evaluation.