Limit Of Sin([x])|x| As X Approaches 0 A Detailed Exploration

Introduction

In the realm of calculus, understanding the behavior of functions as they approach specific points is crucial. This article delves into an intriguing problem involving the limit of a product of two functions, f(x) = sin([x]) and g(x) = |x|, as x approaches 0. Here, [x] denotes the greatest integer function, also known as the floor function, which returns the largest integer less than or equal to x, and |x| represents the absolute value of x. Our goal is to determine the value of $\lim_{x \to 0} f(x)g(x)$, which will provide insights into the interplay between these two distinct functions near the origin.

Understanding the Functions

Before we can evaluate the limit, it's essential to grasp the nature of the functions involved. Let's start with f(x) = sin([x]). The greatest integer function, [x], introduces a step-like behavior to the sine function. For instance, when x is in the interval [0, 1), [x] equals 0, and consequently, sin([x]) = sin(0) = 0. Similarly, when x lies in the interval [-1, 0), [x] equals -1, and sin([x]) = sin(-1), which is a non-zero value. This discrete, step-like nature of [x] significantly impacts the behavior of f(x). Understanding this characteristic is crucial for accurately determining the limit as x approaches 0.

Now, let's consider g(x) = |x|, the absolute value function. This function returns the magnitude of x, irrespective of its sign. In other words, |x| = x for x ≥ 0 and |x| = -x for x < 0. The absolute value function creates a V-shaped graph, with the vertex at the origin (0, 0). As x approaches 0, |x| also approaches 0. This behavior is fundamental to understanding the limit of the product f(x)g(x) as x approaches 0. The interaction between the step-like behavior of f(x) and the linear behavior of g(x) near the origin is what makes this problem particularly interesting.

Evaluating the Limit

To find $\lim_{x \to 0} f(x)g(x)$, we need to analyze the limit from both the left-hand side (as x approaches 0 from negative values) and the right-hand side (as x approaches 0 from positive values). If these one-sided limits exist and are equal, then the overall limit exists and is equal to the one-sided limits. If the one-sided limits are different or do not exist, then the overall limit does not exist. This is a fundamental principle in calculus for determining the existence and value of limits, especially when dealing with functions that have different behaviors depending on the direction of approach.

Right-Hand Limit (x → 0+)

As x approaches 0 from the right (x → 0+), x takes on positive values close to 0. In this case, for 0 < x < 1, the greatest integer function [x] evaluates to 0. Therefore, f(x) = sin([x]) = sin(0) = 0. The absolute value function g(x) = |x| simply equals x for positive x. Thus, the product f(x)g(x) becomes 0 * x = 0. Consequently, the right-hand limit is:

$\lim_{x \to 0^+} f(x)g(x) = \lim_{x \to 0^+} sin([x])|x| = \lim_{x \to 0^+} 0 * x = 0$

This result indicates that as x approaches 0 from the positive side, the product of the two functions converges to 0. This is a crucial piece of information in determining the overall limit. It suggests that the behavior of the functions in the immediate vicinity of 0 from the right does not lead to any divergence or oscillation, but rather a smooth convergence to 0.

Left-Hand Limit (x → 0-)

Now, let's examine the left-hand limit, where x approaches 0 from the left (x → 0-). As x approaches 0 from the negative side, x takes on negative values close to 0. For -1 < x < 0, the greatest integer function [x] evaluates to -1. Therefore, f(x) = sin([x]) = sin(-1), which is a constant value (approximately -0.841). The absolute value function g(x) = |x| equals -x for negative x. Thus, the product f(x)g(x) becomes sin(-1) * (-x). Consequently, the left-hand limit is:

$\lim_{x \to 0^-} f(x)g(x) = \lim_{x \to 0^-} sin([x])|x| = \lim_{x \to 0^-} sin(-1) * (-x) = sin(-1) * \lim_{x \to 0^-} (-x) = sin(-1) * 0 = 0$

This result shows that as x approaches 0 from the negative side, the product of the two functions also converges to 0. Similar to the right-hand limit, there is no divergence or oscillation, but a smooth convergence to 0. This consistency between the left-hand and right-hand limits is a strong indication that the overall limit exists and is equal to 0.

Comparing the Limits

We have determined that the right-hand limit $\lim_{x \to 0^+} f(x)g(x) = 0$ and the left-hand limit $\lim_{x \to 0^-} f(x)g(x) = 0$. Since both one-sided limits exist and are equal, we can conclude that the overall limit exists and is equal to their common value.

Conclusion

Based on our analysis of the one-sided limits, we can definitively state that:

$\lim_{x \to 0} f(x)g(x) = 0$

This result highlights the importance of considering one-sided limits when dealing with functions that may behave differently as they approach a point from different directions. In this case, the interplay between the greatest integer function within the sine function and the absolute value function leads to a convergent limit of 0 as x approaches 0. This exploration not only provides a solution to the specific problem but also underscores the broader principles of limit evaluation in calculus.

Final Answer

The correct answer is (b) 0.