Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow 3^{-}}(x-[x])$$

Step-by-Step Solution

Answer

The left-hand limit as \(x\) approaches 3 is 1.

Step 1: Review the Problem

The problem asks us to find the left-hand limit of the function \(x - [x]\) as \(x\) approaches 3 from the left. Here, \([x]\) represents the greatest integer less than or equal to \(x\) (the floor function). This function calculates the fractional part of \(x\).

Step 2: Understanding the Function

For any real number \(x\), the function \(x - [x]\) gives the fractional part of \(x\), which is always in the interval [0,1). The fractional part is 0 when \(x\) is an integer.

Step 3: Find the Value Approaching 3 from Left

As \(x\) approaches 3 from the left (i.e., \(x = 3^-)\), \([x]\) will be 2 because it takes the greatest integer less than \(x\). Thus, for values close to but less than 3, \(x - [x] = x - 2\).

Step 4: Calculate the Limit

As \(x\) approaches 3 from the left, \(x - 2\) approaches 3 - 2, which is 1. Therefore, \(\lim_{x \to 3^-} (x - [x]) = 1\).

Key Concepts

Left-Hand Limit

The left-hand limit refers to the limit of a function as the variable approaches a particular value from the left side. In mathematical notation, this is expressed as \( \lim_{x \to a^-} f(x) \). Here, \( x \to a^- \) signifies that \( x \) is approaching the number \( a \) from values less than \( a \). It is a crucial concept because it helps us understand the behavior of functions from one specific direction.When we look at the left-hand limit, we often are interested in cases where the limit could be different depending on the direction from which it is approached. If a function is not continuous, the left-hand limit may not be equal to the right-hand limit. In the context of our problem, as \( x \) approaches 3 from the left, we consider values like 2.9, 2.99, or 2.999, which allow us to capture the behavior of the function \( x - [x] \) in this region.

Right-Hand Limit

The right-hand limit involves evaluating a function as the variable approaches a particular number from the right side. This is denoted by \( \lim_{x \to a^+} f(x) \). Similarly to the left-hand limit, the right-hand limit examines the function's behavior, but for values of \( x \) that are just greater than a specific number \( a \).Understanding the right-hand limit is essential in determining the full picture of a function's behavior at a particular point, as it complements the left-hand limit. For continuous functions, the left-hand and right-hand limits are the same. However, for functions with jumps or other discontinuities, these limits can differ. In many problems, calculating the right-hand limit can offer deeper insights, as it might indicate asymptotic behavior or exponentiation behaviors from opposing directions.

Floor Function

The floor function, denoted by \( [x] \), represents the greatest integer less than or equal to \( x \). It essentially "floors" the value, or rounds it down toward negative infinity to the nearest whole number. For instance:

• If \( x = 3.7 \), then \( [x] = 3 \).
• If \( x = -1.2 \), then \( [x] = -2 \).
• If \( x = 4 \), then \( [x] = 4 \).

The floor function is critical in problems involving integer rounding and in defining other functions, like the fractional part function. In the original exercise, it plays a key role by determining that for a value like 2.9 approaching 3 from the left, the function value \([x]\) would be 2, as it is the greatest integer less than \( x \).

Fractional Part of a Number

The fractional part of a number \( x \) can be understood as the part of \( x \) that remains after subtracting its integer part (which is derived using the floor function). It is denoted by \( x - [x] \). This fractional part always lies in the interval \([0, 1)\), meaning it includes zero but never reaches one.Some examples include:

• For \( x = 2.75 \), the fractional part is \( 2.75 - 2 = 0.75 \).
• For \( x = 5.33 \), the fractional part is \( 5.33 - 5 = 0.33 \).
• For \( x = 7 \), the fractional part is \( 7 - 7 = 0 \).

In examining the original problem, we evaluated the fractional part of a number as \( x \) approached 3 from the left. This forms the function \( (x - [x]) \), crucial for understanding the behavior of the function at non-integer values of \( x \), particularly those just below integer thresholds.