Problem 23 - 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 \( \frac{3}{4} \).

Step 1: Recognize the Form of the Limit

Examine the given limit: \( \lim_{y \rightarrow 0} \frac{\sin 3y}{4y} \). Note that it resembles \( \frac{0}{0} \) form because as \( y \rightarrow 0 \), both the numerator \( \sin 3y \rightarrow 0 \) and the denominator \( 4y \rightarrow 0 \) approach zero.

Step 2: Apply the Sine Limit Identity

Recall the fundamental trigonometric limit: \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \). To use this identity, rewrite the expression to closely resemble this form. Recognize that \( \sin 3y \) can be rewritten as \( \frac{\sin 3y}{3y} \cdot \frac{3y}{4y} \).

Step 3: Rewrite the Expression to Utilize the Identity

Rewrite the given expression as \( \frac{\sin 3y}{3y} \cdot \frac{3}{4} \). The limit expression now becomes \( \lim_{y \rightarrow 0} \left( \frac{\sin 3y}{3y} \right) \cdot \frac{3}{4} \).

Step 4: Evaluate the Limits

Evaluate the two individual limits separately. First, apply the identity \( \lim_{y \rightarrow 0} \frac{\sin 3y}{3y} = 1 \). The remaining constant \( \frac{3}{4} \) can be taken out since constants do not affect limits, so the overall expression simplifies to \( 1 \cdot \frac{3}{4} \).

Step 5: Conclude the Solution

Multiply the results from Step 4: \( 1 \cdot \frac{3}{4} = \frac{3}{4} \). Hence, the value of the limit is \( \frac{3}{4} \).

Key Concepts

trigonometric limits

Trigonometric limits are an essential part of calculus and often appear in problems involving trigonometric functions such as sine, cosine, and tangent. One of the most important trigonometric limit identities is the limit of sin(x)/x as x approaches zero, which is commonly used to evaluate complicated limits that might initially present as indeterminate forms like 0/0. In our exercise, we dealt with the limit \( \lim_{y \rightarrow 0} \frac{\sin 3y}{4y} \). Since both the numerator and the denominator approach zero, we can apply the basic trigonometric limit identity: - \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \). Recognizing this common form is key to simplifying and solving problems. By subtly modifying the expression, such as factoring, to fit this identity, we can solve many limits involving trigonometric functions effectively. For our problem, after rewriting \( \sin 3y \) as \( \frac{\sin 3y}{3y} \cdot \frac{3}{4} \), we can confidently apply the trigonometric limit identity, resulting in a simpler and solvable expression.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful technique in calculus for evaluating limits that initially present as indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). This rule states that the limit of a ratio of two functions can be found by differentiating the numerator and the denominator, provided the original limit is indeed indeterminate. In the case of our exercise, \( \lim_{y \rightarrow 0} \frac{\sin 3y}{4y} \) could have been an indeterminate form as both the numerator and the denominator approach zero. If rewriting using trigonometric identities didn't work, L'Hôpital’s Rule would allow evaluation by differentiating the sine and linear terms. Using this method, you would differentiate: - numerator \( \sin 3y \) which becomes \( 3\cos 3y \), - denominator 4y which becomes 4. If the rewritten limit continues to be indeterminate, L'Hôpital's Rule can be reapplied. However, for our problem, applying trigonometric limits resulted in quicker and more intuitive evaluation.

limit evaluation techniques

Evaluating limits can require a few different techniques. Here are some most commonly used strategies, which are essential in handling calculus problems efficiently:

• **Direct Substitution:** Simply substitute the value to which the variable is approaching into the function, if it yields a determinate result.


• **Factoring:** To cancel terms so that indeterminate forms are resolved.


• **Rationalizing:** Multiplying by a conjugate can remove square roots, making evaluation possible.


• **Using known Trigonometric Limits:** Applying identities like \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \) and the similar one for cosine can greatly simplify expressions involving trigonometric functions.


• **L'Hôpital's Rule:** Differentiate the numerator and denominator when directly resulting in an indeterminate form.

In our problem, we applied the known limit of \( \frac{\sin x}{x} \) to elegantly resolve what might have looked complex at first glance. Finding the simplest and most suitable method often leads to clearer insights and correct answers more quickly, an essential aspect of mastering limits and calculus.