Evaluating Limit Of Sin(2x)/x As X Approaches Infinity

In the realm of calculus, evaluating limits is a fundamental concept. Limits help us understand the behavior of functions as their input approaches a specific value, including infinity. In this article, we will delve into the limit of the function f(x) = sin(2x) / x as x approaches infinity. This limit problem combines trigonometric functions with rational functions, presenting an interesting challenge and showcasing important principles of limit evaluation. We will explore the properties of sine function, the concept of boundedness, and the application of the squeeze theorem to rigorously determine the limit.

Before we dive into the limit calculation, let's first understand the behavior of the function f(x) = sin(2x) / x. The function involves two key components: the sine function sin(2x) and the rational function 1/x. The sine function, sin(2x), oscillates between -1 and 1, regardless of the value of x. This is a crucial property, as it makes the sine function bounded. Boundedness means that the function's values are always within a certain range. On the other hand, the rational function 1/x approaches 0 as x approaches infinity. This is because as the denominator grows larger and larger, the overall fraction becomes smaller and smaller, tending towards zero. Understanding these individual behaviors is essential for predicting the limit of their combination.

The oscillating nature of the sine function introduces a unique challenge in evaluating the limit. As x approaches infinity, sin(2x) continues to oscillate between -1 and 1, never settling on a single value. This oscillation prevents us from directly substituting infinity into the function. If we were to ignore the denominator and simply consider sin(2x), the limit would not exist due to the constant oscillation. However, the presence of the denominator x significantly influences the overall behavior of the function. The 1/x term acts as a dampening factor, gradually reducing the impact of the oscillations as x grows larger. This interplay between oscillation and dampening is the key to understanding the limit of the function.

The squeeze theorem, also known as the sandwich theorem, is a powerful tool for evaluating limits when dealing with oscillating functions. The theorem states that if we can find two other functions, g(x) and h(x), that "squeeze" our function f(x) between them, and if the limits of g(x) and h(x) are the same as x approaches a certain value, then the limit of f(x) must also be the same. In mathematical terms, if g(x) ≤ f(x) ≤ h(x) and lim x→a g(x) = lim x→a h(x) = L, then lim x→a f(x) = L. To apply the squeeze theorem to our function f(x) = sin(2x) / x, we need to find suitable bounding functions.

Since we know that the sine function is bounded between -1 and 1, we can write the inequality: -1 ≤ sin(2x) ≤ 1. Now, we divide all parts of the inequality by x, keeping in mind that we are considering the limit as x approaches infinity, so x is positive. This gives us: -1/x ≤ sin(2x) / x ≤ 1/x. We now have our bounding functions: g(x) = -1/x and h(x) = 1/x. These functions squeeze our original function f(x) = sin(2x) / x between them. As x approaches infinity, both g(x) = -1/x and h(x) = 1/x approach 0. This is because the denominator grows without bound, making the fractions tend towards zero.

To formally apply the squeeze theorem, we need to evaluate the limits of our bounding functions as x approaches infinity. The limit of g(x) = -1/x as x approaches infinity is 0: lim x→∞ (-1/x) = 0. Similarly, the limit of h(x) = 1/x as x approaches infinity is also 0: lim x→∞ (1/x) = 0. Since both bounding functions have the same limit, we can now apply the squeeze theorem.

By the squeeze theorem, since -1/x ≤ sin(2x) / x ≤ 1/x and lim x→∞ (-1/x) = lim x→∞ (1/x) = 0, we can conclude that the limit of f(x) = sin(2x) / x as x approaches infinity is also 0: lim x→∞ (sin(2x) / x) = 0. This result demonstrates how the squeeze theorem allows us to handle oscillating functions by bounding them between functions with known limits. The key insight is that the 1/x term effectively dampens the oscillations of sin(2x), causing the function to approach 0 as x becomes very large.

In summary, we have shown that the limit of sin(2x) / x as x approaches infinity is 0. This was achieved by understanding the individual behaviors of the sine function and the rational function, recognizing the challenge posed by the oscillations of the sine function, and applying the squeeze theorem. The squeeze theorem provided a rigorous method for determining the limit by bounding the function between two simpler functions with known limits. This example highlights the power and elegance of the squeeze theorem in evaluating limits involving oscillating functions. Understanding these concepts is crucial for mastering calculus and its applications in various fields.

To deepen your understanding, you can explore similar limit problems involving trigonometric functions and rational functions. Consider functions like cos(x) / x, sin(3x) / x^2, or x * sin(1/x). These examples will further illustrate the application of the squeeze theorem and the interplay between oscillations and dampening factors. Additionally, exploring graphical representations of these functions can provide visual intuition for their limiting behavior. Understanding limits is a cornerstone of calculus, and mastering these techniques will open doors to more advanced concepts and applications.