Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

Many software packages have programs for calculating limits, although you should be warned that they are not infallible. To develop confidence in your program, use it to recalculate some of the limits in Problems 1-28. Then for each of the following, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0}(\sin 2 x) / 4 x $$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{2} \).

Step 1: Identify the Limit Expression

The limit we need to evaluate is \( \lim_{x \rightarrow 0} \frac{\sin 2x}{4x} \). We notice that the expression has a function involving sine, similar to the fundamental limit \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \).

Step 2: Rewrite with Known Limit Formula

To use the known limit formula, rewrite the expression by factoring out constants and aligning it with \( \frac{\sin \, ext{(angle)}}{\text{(same angle)}} \). We factor 2 out from the sine expression: \( \frac{1}{2} \times \frac{\sin 2x}{2x} \).

Step 3: Apply the Fundamental Sine Limit

Using the fundamental limit \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \), apply it to \( \lim_{x \rightarrow 0} \frac{\sin 2x}{2x} \). This gives us \( 1 \).

Step 4: Combine Results

Since \( \lim_{x \rightarrow 0} \frac{\sin 2x}{2x} = 1 \) and we have a factor of \( \frac{1}{2} \) in front, the overall limit is \( \frac{1}{2} \times 1 = \frac{1}{2} \).

Key Concepts

Fundamental Sine Limit

The Fundamental Sine Limit is a crucial building block in calculus, specifically when dealing with trigonometric functions and limits. It plays a significant role in solving problems that involve the sine function. This limit is expressed as \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \). This remarkable limit tells us that as \( x \) approaches zero, the ratio \( \frac{\sin x}{x} \) approaches one.

This concept is powerful because it helps simplify expressions involving sine functions when calculating limits. Since sine is a continuous and smooth function, near zero, \( \sin x \) behaves very similarly to \( x \) itself, leading to the limit approaching 1.

• Understanding this limit can help in evaluating more complex expressions with sine.
• It is often used to manipulate and simplify trigonometric expressions.
• Remember, familiarity with this fundamental limit is a must to progress in calculus.

Limit Evaluation

Limit Evaluation is at the heart of calculus as it explores what a function does as it approaches a particular point. To evaluate limits effectively, especially those involving trigonometric functions, knowledge of certain strategies is essential.

In the case of the expression \( \lim_{x \rightarrow 0} \frac{\sin 2x}{4x} \), you begin by recognizing the structure of the given function and its similarities to known limits and identities.

• Observe the function form, specifically isolating components similar to the fundamental sine limit, \( \frac{\sin x}{x} \).
• Factor out constants if necessary to match the known limit forms, like rewriting \( \frac{\sin 2x}{4x} \) as \( \frac{1}{2} \times \frac{\sin 2x}{2x} \).
• Apply known limits to simplify and evaluate the limit.
• Combine results of the evaluation to find the overall limit result.

By developing these strategies and recognizing these patterns, evaluating limits becomes more straightforward and less susceptible to errors.

Trigonometric Limits

Trigonometric Limits often involve limits with sine, cosine, and other trigonometric functions. Understanding these limits is vital in calculus, particularly in dealing with derivatives and integrals involving trigonometric functions.

In our exercise, we used the trigonometric limit \( \lim_{x \rightarrow 0} \frac{\sin 2x}{2x} \).

• Trigonometric limits often require the application of identities and fundamental limits, like \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \).
• Combining trigonometric limits with algebraic manipulations helps evaluate them effectively.
• Recognize if direct substitution yields an indeterminate form, such as \( \frac{0}{0} \), and apply appropriate techniques to resolve it.
• Frequently, expressions need to be rewritten or reshaped to fit known trigonometric limits for easy evaluation.

Working through these problems requires understanding how trigonometric functions behave near certain points and leveraging known limit-solving techniques.