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Chapter 3: Problem 9

Determine all values of \(\bar{x}\) such that the limit \(\lim _{x \rightarrow \bar{x}}(1+x-[x])\) exists.

Short Answer

Expert verified
The limit exists for all non-integer values of \(\bar{x}\).

Step by step solution

01

Understand the Function Components

The given expression is \(1 + x - [x]\). Here, \([x]\) represents the floor function, which gives the greatest integer less than or equal to \( x \). Our goal is to find values of \(\bar{x}\) for which the limit exists as \(x\) approaches \(\bar{x}\).
02

Analyze the Limit

The expression \(1 + x - [x]\) simplifies to \(1 + \{x\}\), where \(\{x\} = x - [x]\) is the fractional part of \(x\). The function becomes \(1 + \{x\}\), which is a real value between 1 and 2, excluding 2, since \(\{x\}\) ranges from 0 to just less than 1.
03

Determine the Discontinuity Points

The function \(1 + \{x\}\) is continuous for all real \(x\) except at integer points where \(x = n\) (\(n\) is any integer). At these points, \(\lim_{{x \to n^-}} = 2\) and \(\lim_{{x \to n^+}} = 1\); hence, the limit does not exist at integer values.
04

Identify Values for Which Limit Exists

The limit \(\lim_{x \to \bar{x}} (1 + x - [x])\) exists for all non-integer values of \(\bar{x}\), as these are the points where the function is continuous.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit of a Function
In mathematical analysis, the term "limit" describes the behavior of a function as its argument approaches a particular value. When we talk about the limit of a function such as \( \lim_{x \to \bar{x}} (1 + x - [x]) \), we are examining how the function behaves near \( \bar{x} \). Here, the expression inside the limit simplifies to \( 1 + \{x\} \), where \( \{x\} \) represents the fractional part of \( x \).
  • The "limit exists" if we can find a single value that the function approaches as \( x \) gets infinitely close to \( \bar{x} \) from both sides.
  • For the given function, the limit fails to exist wherever there is a jump or discontinuity.
This concept is foundational in calculus and helps in understanding not just continuity and differentiability but also the behavior of sequences and series.
Floor Function
The floor function, denoted by \( [x] \) or sometimes \( \lfloor x \rfloor \), outputs the largest integer smaller than or equal to \( x \). In simpler terms, it "rounds down" to the nearest whole number.
  • For example, \( [3.7] = 3 \) and \( [-2.3] = -3 \).
  • In the context of our function, \( [x] \) represents the integer part of \( x \).
  • This function creates a step-like graph where each step extends from one integer to just before the next one.
The floor function is crucial in our expression as it helps in isolating the fractional part \( \{x\} \), which is used to determine continuity and discontinuity.
Continuity
Continuity at a point means that the graph of a function does not have any breaks, jumps, or holes at that point. For the function \( 1 + \{x\} \), continuity applies for all non-integer values of \( x \).
  • A continuous function at a point \( x = a \) satisfies the condition: \( \lim_{x \to a} f(x) = f(a) \).
  • In simpler terms, you can draw the function without lifting your pencil off the paper at that point.
For our problem, this means that the limit exists for non-integer values because the fractional part of \( x \) changes smoothly between integers without any abrupt jumps.
Discontinuity
Discontinuity in a function is when it has breaks, jumps, or holes, meaning the limit does not exist at those points. For the function \( 1 + \{x\} \), discontinuities occur at integer values of \( x \).
  • Discontinuity occurs because the fractional part \( \{x\} \) resets to zero each time \( x \) reaches an integer number.
  • At an integer \( n \), \( \lim_{{x \to n^-}} \) approaches 2, and \( \lim_{{x \to n^+}} \) approaches 1.
  • Because these two one-sided limits differ, the function is discontinuous at integers.
Understanding discontinuities helps to predict where limits fail to exist and where the function's graph makes abrupt jumps.
Fractional Part of a Number
The fractional part of a number \( x \), denoted as \( \{x\} \), is the non-integer part of the number, computed as \( x - [x] \).
  • It always lies in the interval \( [0, 1) \).
  • For example: if \( x = 3.75 \), then \( [x] = 3 \) and \( \{x\} = 0.75 \).
  • This operation isolates the decimal component of \( x \), which repeats in the same manner between each pair of consecutive integers.
In our function, \( 1 + \{x\} \) can vary between 1 and 2, but never actually reaching 2, due to the nature of the fractional part. This characteristic is key to determining continuity and discontinuity at specific points.

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Most popular questions from this chapter

Give an example of a subset \(D\) of \(\mathbb{R}\) and uniformly continuous functions \(f, g: D \rightarrow \mathbb{R}\) such that \(f g\) is not uniformly conitnuous on \(D\).

Let \(f: D \rightarrow \mathbb{R}\) and let \(\bar{x}\) be a limit point of \(D\). Suppose that $$ |f(x)-f(y)| \leq k|x-y| \text { for all } x, y \in D \backslash\\{\bar{x}\\}, $$ where \(k \geq 0\) is a constant. Prove that \(\lim _{x \rightarrow \bar{x}} f(x)\) exists.

Let \(f, g:[0,1] \rightarrow[0,1]\) be continuous functions such that $$ f(g(x))=g(f(x)) \text { for all } x \in[0,1]. $$ Suppose further that \(f\) is monotone. Prove that there exists \(x_{0} \in[0,1]\) such that $$ f\left(x_{0}\right)=g\left(x_{0}\right)=x_{0}. $$

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Let \(D \subset \mathbb{R}, f: D \rightarrow \mathbb{R},\) and \(\bar{x}\) be a limit point of \(D .\) Prove that \(\liminf _{x \rightarrow \bar{x}} f(x) \leq \limsup _{x \rightarrow \bar{x}} f(x)\).
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