Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of \(f(x)=x-[x]\); then find each of the following or state that it does not exist. (a) \(f(0)\) (b) \(\lim _{x \rightarrow 0} f(x)\) (c) \(\lim _{x \rightarrow 0^{-}} f(x)\) (d) \(\lim _{x \rightarrow 1 / 2} f(x)\)

Step-by-Step Solution

Answer

(a) 0 (b) 0 (c) 1 (d) 0.5

Step 1: Understanding the Function

The function given is \( f(x) = x - [x] \), where \([x]\) is the greatest integer function or floor function. It represents the greatest integer less than or equal to \( x \). This function essentially yields the fractional part of \( x \). For instance, if \( x = 2.3 \), \([x] = 2\) and \( f(x) = 2.3 - 2 = 0.3 \). This function will always yield a value between \(0\) and \(1\), for any \(x\).

Step 2: Sketching the Graph

To sketch the graph of \( f(x) = x - [x] \), note that for any integer \( n \), the function will reset to 0. Between any two consecutive integers, say \( n \) and \( n+1 \), the graph is a straight line from \((n, 0)\) to \((n+1, 1)\) but does not include \(1\) at \(n+1\). The function is discontinuous at each integer point because it jumps back to 0.

Step 3: Evaluating \(f(0)\)

Evaluate \( f(0) \). Here, \([0] = 0\), so \( f(0) = 0 - 0 = 0 \).

Step 4: Limit as \( x \to 0 \)

Find \( \lim_{x \to 0} f(x) \). Approaching 0 from both the left and right, \( f(x) = x - 0 = x \) near zero, thus the limit is \( \lim_{x \to 0} x = 0 \).

Step 5: Left-hand Limit as \( x \to 0^- \)

Find \( \lim_{x \to 0^-} f(x) \). As x approaches 0 from the left (negative side), \( f(x) = x - (-1) = x + 1 \) as \([x] = -1\). Therefore, \( \lim_{x \to 0^-} f(x) = 0 + 1 = 1 \).

Step 6: Limit as \( x \to \frac{1}{2} \)

Find \( \lim_{x \to \frac{1}{2}} f(x) \). As \( x \) approaches \( \frac{1}{2} \), \([x] = 0\), so \( f(x) = x - 0 = x \). Therefore, the limit is \( \frac{1}{2} \).

Key Concepts

Greatest Integer Function

The greatest integer function, commonly known as the floor function, is represented as \([x]\). This function takes an input \( x \) and returns the greatest integer less than or equal to \( x \). In other words, it "rounds down" \( x \) to the nearest integer. \([x]\) is particularly interesting in problems involving piecewise and discontinuous functions because of its step-like behavior.

For example:

• If \( x = 3.7 \), then \([x] = 3\).
• If \( x = -2.3 \), then \([x] = -3\).
• If \( x = 5 \), then \([x] = 5\).

The greatest integer function creates a unique graph that consists of horizontal steps, with breaks at every integer point. This characteristic affects the behavior of functions that use it, especially in determining limits and continuity.

Limit of a Function

The limit of a function as \( x \) approaches a certain value is a fundamental concept in calculus. It describes the behavior of the function near that point. Knowing whether limits exist or not is crucial for understanding how the function behaves as it nears a particular value, but not necessarily at that exact value.

For \([f(x) = x-[x]]\), the greatest integer function influences the limits at integer values. Particularly near zero, you have different behavior on the left (approaching from negative values) and the right (approaching from positive values). That's because \( f(x) = x-[x] \) changes how we consider limits at points of discontinuity.

For example:

• Approaching \( x = 0 \) from the right, \( f(x) = x \) results in a limit of \( 0 \).
• Approaching \( x = 0 \) from the left however, yields a different behavior: \( f(x) = x + 1 \), resulting in a limit of \( 1 \).

These different directional limits highlight the function's discontinuity, a crucial insight when working with piecewise or step functions.

Piecewise Functions

Piecewise functions are defined by multiple sub-functions, each applying to a specific portion of the domain. They are a great example where calculus meets real-world applications because real-world phenomena often don't follow a single simple equation. These functions allow different formulas or rules based on varying conditions.

In the case of the function \( f(x) = x - [x] \), consider it as a piecewise combination of linear segments:

• For any interval \( n \leq x < n+1\), \( f(x) = x - n \). Here, each piece is linear, resetting at every integer \( n \).

Piecewise functions may have discontinuities, meaning they change suddenly at certain points, and these traits make them intriguing and useful in modeling scenarios with distinct bounded phases.

Continuity and Discontinuity

Continuity in a function means there's no interruption in its value as it progresses along its domain. When a function is continuous at a point, its limit approaches and equals the function's value at that point. Discontinuity involves interruptions where the function jumps, breaks, or doesn't exist.

The function \( f(x) = x - [x] \) clearly illustrates this concept by being discontinuous at integer points. At these locations, the value of the function jumps, as indicated when sketching its graph. At each integer \( n \):

• Right before \( n \), \( f(x) \) approaches \( 1 \).
• Exactly at \( n \), \( f(x) \) resets to \( 0 \).

This discontinuity is the core reason why limits from the left and right at these points do not match, reflecting breaks in the graph. Understanding these concepts helps clarify why functions behave unpredictably at certain points, a cornerstone in advanced calculus problems.