Question

When \(x\) is near \(0, \frac{\sin x}{x}\) is near what value?

Short answer

The value of \( \frac{\sin x}{x} \) approaches 1 as \( x \to 0 \).

Step-by-step solution

Understanding the Problem

We are asked to find the value of \( \frac{\sin x}{x} \) as \( x \to 0 \). This is a classic limit problem in calculus.

Utilizing the Sine Limit Identity

One of the standard limits in calculus is \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). According to this identity, as the value of \( x \) approaches zero, the value of \( \frac{\sin x}{x} \) approaches 1.

Analyzing the Behavior of Sine Around Zero

The function \( \sin x \) behaves such that when \( x \) is small, \( \sin x \approx x \). So, \( \frac{\sin x}{x} \approx \frac{x}{x} = 1 \). This confirms the identity that as \( x \) approaches zero, \( \frac{\sin x}{x} \) nears 1.

Conclusion Using Calculus Concepts

Using the definition of limits and the behavior of trigonometric functions, we conclude that \( \frac{\sin x}{x} \) approaches 1 as \( x \) approaches 0.

Key concepts

Limits

Limits are a fundamental concept in calculus that refers to the value that a function approaches as the input approaches some particular point. This allows us to understand the behavior of functions near specific values, even if the function isn't defined at that point.
For example, consider the expression \( \lim_{x \to 0} \frac{\sin x}{x} \). This notation asks us to evaluate what happens to \( \frac{\sin x}{x} \) as \( x \) approaches 0. Despite \( \frac{\sin x}{x} \) not being defined at exactly \( x = 0 \) (since it would involve division by zero), the limit tells us what value \( \frac{\sin x}{x} \) gets closer and closer to as \( x \) nears 0.

Trigonometric Functions

Trigonometric functions are functions that relate the angles of triangles to the lengths of their sides. They are periodic and fundamental to studying oscillatory motions such as sound and light waves.
Common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions have unique properties and are critical in understanding various real-world and theoretical problems within mathematics and physics.
In calculus, these functions often appear in problems involving limits, derivatives, and integrals, allowing us to explore more complex mathematical applications.

Sine Function

The sine function, expressed as \( \sin(x) \), is one of the primary trigonometric functions. It calculates the y-coordinate of a point on the unit circle that an angle \( x \) subtends from the positive x-axis.
Its value oscillates between -1 and 1 as \( x \) progresses around the circle. This periodic nature means \( \sin(x) \) repeats its values in regular intervals, specifically over every \( 2\pi \) radians or 360 degrees.
In the context of the original exercise, the behavior of the sine function when \( x \) is close to zero is particularly important. Under these conditions, \( \sin(x) \approx x \), a crucial approximation that simplifies evaluating limits involving \( \sin(x) \) near 0.

Limit Identity

The limit identity \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) is a well-known result in calculus. It confirms that as \( x \) approaches zero, the ratio \( \frac{\sin x}{x} \) approaches the value of 1.
This identity is particularly useful because it provides a simple method to analyze the behavior of sine at small values. It's derived from the fact that near zero, the sine function behaves very similarly to its angle in radians. Therefore, \( \sin(x) \) and \( x \) become almost indistinguishable.
This foundational identity is widely used in calculus courses to solve more complex limit problems involving trigonometric functions. Recognizing and applying this identity can greatly simplify analyzing and solving limits.