Problem 24 - Calculus Early Transcendentals

Question

Evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\sin ^{2} 3 x}{x^{2}}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit of the function $$\frac{\sin ^{2} 3 x}{x^{2}}$$ as \(x\) approaches \(0\). Answer: When \(x\) approaches \(0\), the limit of the function is $$1$$.

Step 1: Write down the given function

$$\lim _{x \rightarrow 0} \frac{\sin ^{2} 3 x}{x^{2}}$$

Step 2: Break apart \(\sin^{2} 3x\) term

Apply the rule \(\sin^2 u = (\sin u)^2\) and break apart the term \(\sin^2 3x\) as \((\sin 3x)^2\): $$\lim _{x \rightarrow 0} \frac{(\sin 3x)^{2}}{x^{2}}$$

Step 3: Introduce a term \(3\) in the numerator and denominator

Divide and multiply by \(3^2\) to introduce a term similar to the standard limit property, resulting in: $$\lim _{x \rightarrow 0} \frac{3^2(\sin 3x)^{2}}{3^2x^{2}}$$ This can be further simplified to: $$ \lim _{x \rightarrow 0} \left(\frac{3\sin 3x}{3x}\right)^{2} $$

Step 4: Apply the standard limit property

Now that the expression inside the parentheses is similar to the \(\lim_{x\rightarrow 0} \frac{\sin x}{x}\), substitute the \(1\) for the standard limit property in the expression: $$\left(\lim_{x\rightarrow 0} \frac{3\sin 3x}{3x}\right)^2$$ The expression becomes: $$(1)^2$$

Step 5: Final result

When \(x\) approaches \(0\), the limit of the function becomes: $$\lim _{x \rightarrow 0} \frac{\sin ^{2} 3 x}{x^{2}} = 1^2 = 1$$

Key Concepts

Sine function

The sine function, often denoted by \( \sin(x) \), is a fundamental trigonometric function, commonly used in calculus and various applications like physics and engineering. It stems from the unit circle, where for an angle \( x \), \( \sin(x) \) is the y-coordinate of the point on the circle. The sine function is periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) units. This periodic nature can be particularly useful when analyzing oscillating systems or waves.

• It is an odd function: \( \sin(-x) = -\sin(x) \)
• The range of the sine function is [-1, 1]
• Its graph is a smooth, continuous wave-like curve

When dealing with derivatives and limits, understanding how to manipulate equations involving the sine function is crucial. In limit problems, recognizing patterns that involve the sine function can provide a pathway to simplification, such as reducing it to well-known standards or identities.

Standard limit property

Within calculus, several fundamental limits are memorized or derived due to their frequent recurrence. One such property relates to the sine function:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
This property is significant because it simplifies calculations and serves as a building block for evaluating other complex limits. The standard limit is essential when \( x \to 0 \), helping in the "squeezing" technique or in identifying limit behavior close to zero.

• Applies specifically when the angle (in radians) approaches zero
• Facilitates simplifying expressions where sine is used in a ratio with its input
• Can be squared or manipulated as needed, provided when the same variable approaches zero

Recognizing parts of a limit that can adopt this form is crucial when solving calculus problems. It makes approaching complex problems manageable and more intuitive.

Limit evaluation techniques

Evaluating limits often requires multiple strategies and techniques, especially when dealing with trigonometric functions like sine. Here are some common methods to tackle these problems:

• Direct Substitution: Attempt initially by substituting the approaching value directly into the function. If it leads to an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), other methods are required.
• Algebraic Manipulation: Simplify the function using algebraic identities or expressions. As seen in our solution, observing and manipulating into a recognized form is essential.
• Factoring: Factor both the numerator and denominator where possible and eliminate common factors.
• Substitution: Sometimes substituting variables can transform the complex limit into a simpler form.
• Standard Limits: Apply known limits, such as \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \), as seen in the problem solution above.

Successful application of these techniques depends on understanding underlying functions and recognizing patterns. Each problem may require a unique combination of these approaches for a solution.