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. $$ \left.\lim _{x \rightarrow 3^{-}}(x-\llbracket x]\right) $$

Step-by-Step Solution

Answer

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

Step 1: Understand the function involved

The expression inside the limit is \( x - \lfloor x \rfloor \), where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \). This function represents the fractional part of \( x \).

Step 2: Approach from the Left

Since we are dealing with \( x \to 3^- \), it means \( x \) is approaching 3 from the left. Just before \( x \) reaches 3, it is just slightly less than 3. In this interval, \( x < 3 \) and \( \lfloor x \rfloor = 2 \).

Step 3: Substitute to Find Limit

Substitute into the expression: \( x - \lfloor x \rfloor = x - 2 \). As \( x \to 3^- \), this expression becomes \( 3 - 2 = 1 \).

Step 4: Conclusion

Since the value of the expression as \( x \to 3^- \) consistently approaches 1, the left-hand limit exists and equals 1.

Key Concepts

Left-Hand Limit

In calculus, the left-hand limit refers to the value that a function approaches as the input approaches a specified point from the left side. Imagine a number line where you move towards a specific point from the lower number side. For example, if you want to find the left-hand limit as \( x \) approaches 3, you would consider values slightly less than 3 like 2.9 or 2.99.

The left-hand limit is noted as \( \lim_{x \to c^-} f(x) \), which means you are considering values of \( x \) approaching \( c \) from the left. Here’s how you find it for a function:

• Identify the function and the point you are approaching from the left.
• Consider values just to the left of the point, substitute into the function.
• Observe the output to determine what value it approaches.

This helps us understand the behavior of the function as the input gets very close to that limit value but never actually reaches it. So, in our original exercise, we found that as \( x \) approaches 3 from the left, \( x - \lfloor x \rfloor \) approached 1, making the left-hand limit equal to 1.

Right-Hand Limit

The right-hand limit is similar to the left-hand limit, but this time we approach our point from the right side of the number line. When you find the right-hand limit, you consider values slightly greater than the point of interest.

Mathematically, this is written as \( \lim_{x \to c^+} f(x) \). So if you’re finding the right-hand limit as \( x \) approaches 3, you would investigate \( x \) values such as 3.01 or 3.1. The steps are quite analogous to finding the left-hand limit:

• Choose the function and the targeted point from which you're approaching from the right.
• Plug slightly greater values into the function.
• Determine the trend of these calculated outputs to identify the approaching value.

This is a crucial concept as it assists in understanding continuity and the complete behavior of functions around a point. Ensuring that both left-hand and right-hand limits exist and are equal determines if a function is continuous at that specific point.

Greatest Integer Function

The greatest integer function, also known as the floor function and represented by \( \lfloor x \rfloor \), maps a real number to the largest integer less than or equal to that number. It effectively "rounds down" to the nearest whole number.

Consider the function \( \lfloor x \rfloor \) when \( x = 2.7 \). The greatest integer less than or equal to 2.7 is 2, so \( \lfloor 2.7 \rfloor = 2 \). Here’s a breakdown of its properties:

• Non-continuous: The function creates steps, meaning it jumps at whole numbers.
• Piecewise constant: Within any interval between two integers, the function remains constant.

This function is particularly useful in the original exercise, as it defines the integer portion of \( x \) when examining expressions like \( x - \lfloor x \rfloor \). By understanding it, you can comprehend the fundamental role it plays in calculating limits and piecewise functions.