Problem 8 - Calculus Early Transcendentals

Question

Find the limits. $$ \lim _{x \rightarrow 0} \frac{\sin 2 x}{\sin 5 x} $$

Step-by-Step Solution

Answer

The limit is \( \frac{2}{5} \).

Step 1: Understand the limit problem

We need to find the limit of the expression \( \lim _{x \rightarrow 0} \frac{\sin 2x}{\sin 5x} \). This is an indeterminate form \( \frac{0}{0} \), which needs to be simplified or evaluated using a known limit involving sine.

Step 2: Use the limit identity

A useful identity when working with sine limits is \( \lim_{u \to 0} \frac{\sin u}{u} = 1 \). To use this identity, we will manipulate the expression so that it resembles this form.

Step 3: Rewrite the expression

We need to rewrite \( \frac{\sin 2x}{\sin 5x} \) by introducing factors that will allow us to apply the identity. Rewrite this as: \( \frac{2x}{5x} \times \frac{\sin 2x}{2x} \times \frac{5x}{\sin 5x} \).

Step 4: Apply the known sine limit

Now, apply the known limit to each part: \( \lim_{x \to 0} \frac{\sin 2x}{2x} = 1 \) and \( \lim_{x \to 0} \frac{5x}{\sin 5x} = 1 \). So the entire expression simplifies to \( \frac{2}{5} \times 1 \times 1 = \frac{2}{5} \).

Step 5: Calculate the final limit

Multiply the parts: \( \frac{2}{5} \times 1 \times 1 = \frac{2}{5} \). Thus, the original limit evaluates to \( \frac{2}{5} \).

Key Concepts

Trigonometric Limits

In calculus, limits involving trigonometric functions often require specific techniques as these limits can result in indeterminate forms. A widely used trigonometric limit is \( \lim_{u \to 0} \frac{\sin u}{u} = 1 \). This identity is vital when solving limits that involve sine or cosine functions, particularly when the input approaches zero.

Understanding this limit identity is crucial because it allows for the simplification of expressions that initially look complicated. With trigonometric limits, it is common to rearrange or manipulate the mathematical expression to utilize such known limits effectively. This is often done by introducing multipliers that match the argument of the sine functions with the denominator or vice versa, enabling us to apply the limit identity accurately.

The process we use in trigonometric limits also serves as a foundation for understanding more complex calculations involving trigonometric functions in calculus, as these principles extend to derivatives and integrals.

L'Hopital's Rule

L'Hopital's Rule is a powerful tool in calculus, primarily used for finding limits that yield indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The rule states that if \( \lim_{x \to a} \frac{f(x)}{g(x)} \) is indeterminate, then this limit can be evaluated as \( \lim_{x \to a} \frac{f'(x)}{g'(x)} \), given these derivatives exist around the point.

To apply L'Hopital's Rule effectively, ensure that both the numerator and denominator approach zero or infinity as \( x \to a \). In our trigonometric example, the expression originally created an indeterminate form \( \frac{0}{0} \), making L'Hopital’s Rule initially applicable. However, given the simplicity of sine and cosine, using known limits can sometimes result in a more straightforward solution.

To decide when to use L'Hopital's Rule, consider whether the derivatives involved will simplify the expression. If they make computations complex or unnecessary, consider using limit identities or algebraic manipulation instead.

Indeterminate Forms

Indeterminate forms arise in calculus when evaluating limits, leading to uncertain or undefined results, such as \( \frac{0}{0} \), \( \infty - \infty \), or \( 0 \times \infty \). The central idea is that these forms do not provide enough information to determine a precise limit without further manipulation or application of calculus techniques.

In the original exercise, the expression \( \lim_{x \to 0} \frac{\sin 2x}{\sin 5x} \) initially results in an indeterminate form of \( \frac{0}{0} \). This is typical of many trigonometric expressions where both numerator and denominator approach zero at the zero point. Indeterminate forms require additional techniques, such as algebraic manipulation, applying trigonometric identities, or using L'Hopital's Rule to resolve the limit precisely.

Recognizing these forms early is critical, allowing you to apply appropriate methods to simplify and evaluate limits accurately. Understanding various approaches to tackling indeterminate forms is a valuable skill, providing clarity and efficiency in solving complex calculus problems.