Problem 8 - Calculus for Biology and Medicine

Question

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{\sin ^{2}(2 x)}{x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Recognize the Limit Form

The given limit is \( \lim_{x \to 0} \frac{\sin^2(2x)}{x} \). This does not immediately fit the standard limit form \( \frac{\sin(x)}{x} \to 1 \) as \( x \to 0 \). However, we can reformulate it in the subsequent steps.

Step 2: Apply Double Angle Identity and Substitution

Since \( \sin^2(2x) = (\sin(2x))^2 \), rewrite the expression as:\[ \lim_{x \to 0} \frac{(\sin(2x))^2}{x} = \lim_{x \to 0} \frac{\sin(2x)}{x} \cdot \sin(2x) \]Next, substitute \( u = 2x \). Then, as \( x \to 0 \), \( u \to 0 \). Thus, we can rewrite the limit:\[ \lim_{u \to 0} \frac{\sin(u)}{u/2} \cdot \sin(u) = \lim_{u \to 0} 2 \cdot \frac{\sin(u)}{u} \cdot \sin(u) \]

Step 3: Apply Standard Limit for Sine Function

We know that \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \). Thus, applying this identity, we have:\[ \lim_{u \to 0} 2 \cdot \frac{\sin(u)}{u} \cdot \sin(u) = \lim_{u \to 0} 2 \cdot 1 \cdot \sin(u) \]

Step 4: Evaluate \( \sin(u) \) at Zero

Since \( \sin(u) \to 0 \) as \( u \to 0 \), we have:\[ \lim_{u \to 0} 2 \cdot \sin(u) = 2 \cdot 0 = 0 \]

Step 5: Final Limit Value

Thus, the final answer for the limit is:\[ \lim_{x \to 0} \frac{\sin^2(2x)}{x} = 0 \]

Key Concepts

Understanding the Sine Function

The sine function, denoted as \(\sin(x)\), is one of the basic trigonometric functions. It describes the vertical position of a point moving around a circle of radius one. Trigonometric functions like sine often arise in the context of waves and oscillations.
The sine of an angle in a right triangle is defined as the ratio of the opposite side to the hypotenuse. For angles measured in radians, as \(x\) approaches zero, the value of \(\sin(x)\) is very close to the value of \(x\) itself. This property is crucial when evaluating limits, especially when dealing with small-angle approximations.

• Range: The sine function varies between -1 and 1.
• Periodicity: It is periodic with a period of \(2\pi\).
• Small-angle approximation: For small angles, \(\sin(x) \approx x\).

Understanding these properties can greatly aid in solving limits involving the sine function, especially as \(x\) approaches a very small value, like zero.

Double Angle Identity in Trigonometry

The double angle identity is one of the many useful identities in trigonometry. It involves expressing trigonometric functions of angles as functions of twice those angles. Specifically, the identity for sine is: \(\sin(2x) = 2 \sin(x)\cos(x)\).
This identity helps simplify problems and evaluate expressions more easily. In limit problems, identifying opportunities to apply known identities can transform the problem into something more familiar. For example, rewriting \(\sin^2(2x)\) as \((\sin(2x))^2\), and then using substitution, allows us to work with expressions in simpler forms.

• This identity is especially useful when dealing with angles in different forms, like doubling or halving them.
• It allows conversion of products of trigonometric functions into sums, which are easier to handle.

Such strategic manipulations are key when evaluating limits involving trigonometric functions.

Techniques for Limit Evaluation

Evaluating limits, especially those involving trigonometric functions, requires tactful simplification using algebraic identities and known theorems.
When dealing with trigonometric limits like \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\), recognizing patterns and employing techniques such as substitution or trigonometric identities is essential. Breaking down complex expressions into simpler, solvable parts can make a problem less daunting.

• Substitution: Changing variables can sometimes recast the limit into a standard form.
• Identity application: Using known trigonometric identities to rewrite expressions is often necessary.
• Small-angle theorems: The behavior of sine functions near zero is leveraged in limit evaluation.
• Simplification: Rewriting functions to match known limit forms helps us apply the limit theorems effectively.

Mastering these techniques fine-tunes your problem-solving skills, making seemingly complex trigonometric limits approachable and solvable.