Problem 115
Question
Let \(f, g\) and \(h\) be real valued functions defined on an interval \(I \subseteq R\) except possibly for some point \(c\) such that $$ \lim _{x \rightarrow c} f(x)=l=\lim _{x \rightarrow c} h(x) $$ and, \(f(x) \leq g(x) \leq h(x), \forall x \in I\). Then, \(\lim _{x \rightarrow c} g(x)=l\). \(\lim _{n \rightarrow \infty} \frac{\\{x\\}+\\{2 x\\}+\\{3 x\\}+\ldots+\\{n x\\}}{n^{2}}\) where \(\\{x\\}=x-[x]\) denotes the fractional part of \(x\), is (A) 1 (B) 0 (C) \(\frac{1}{2}\) (D) None of these
Step-by-Step Solution
Verified Answer
The limit is 0, so the correct answer is (B).
1Step 1: Understanding the Fractional Part
The notation \( \{x\} = x - \lfloor x \rfloor \) represents the fractional part of a number \(x\). The fractional part of \(x\) is always a number between 0 (inclusive) and 1 (exclusive).
2Step 2: The Expression Inside the Limit
Consider the expression \( \lim_{n \rightarrow \infty} \frac{\{x\} + \{2x\} + \{3x\} + \, \ldots \, + \{nx\}}{n^2} \). We note that each term \( \{kx\} \) for \( k = 1, 2, \, \ldots \, , n \) is bounded by 0 and 1. Thus the entire numerator is a sum of \( n \) terms each of which is at most 1.
3Step 3: Bounding the Sum
Since each term \( \{kx\} \) is at most 1, the maximum possible value of the sum \( \{x\} + \{2x\} + \{3x\} + \, \ldots \, + \{nx\} \) is \(n\). Hence, we have the inequality \(0 \leq \{x\} + \{2x\} + \{3x\} + \, \ldots \, + \{nx\} \leq n\).
4Step 4: Evaluating the Limit
We want to calculate the limit \( \lim_{n \rightarrow \infty} \frac{\{x\} + \{2x\} + \{3x\} + \, \ldots \, + \{nx\}}{n^2} \). Using the bounding from the earlier step, we get \( 0 \leq \frac{\{x\} + \{2x\} + \{3x\} + \, \ldots \, + \{nx\}}{n^2} \leq \frac{n}{n^2} = \frac{1}{n} \).
5Step 5: Applying the Squeeze Theorem
Since \( \frac{1}{n} \to 0 \) as \( n \to \infty \) and the fraction is squeezed between 0 and \( \frac{1}{n} \), by the Squeeze Theorem, \( \lim_{n \rightarrow \infty} \frac{\{x\} + \{2x\} + \{3x\} + \, \ldots \, + \{nx\}}{n^2} = 0 \).
Key Concepts
Fractional PartLimit of a SequenceBounded Functions
Fractional Part
When you encounter the notation \(\{x\} = x - \lfloor x \rfloor\), it refers to the fractional part of a number \(x\). This concept is pretty straightforward. Essentially, it represents the difference between a number and its greatest integer less than or equal to that number (also called the integer part or floor function).
For example, for the number 5.6, the fractional part is 0.6 since \(\lfloor 5.6 \rfloor = 5\). An important aspect of the fractional part is that it is always within the range from 0 (inclusive) to 1 (exclusive).
This means that no matter what number you start with, the fractional part is a number that's never quite 1, or negative, truly encapsulating the decimal elements of any given number.
For example, for the number 5.6, the fractional part is 0.6 since \(\lfloor 5.6 \rfloor = 5\). An important aspect of the fractional part is that it is always within the range from 0 (inclusive) to 1 (exclusive).
This means that no matter what number you start with, the fractional part is a number that's never quite 1, or negative, truly encapsulating the decimal elements of any given number.
Limit of a Sequence
The concept of finding the limit of a sequence is crucial for understanding behavior as \(n\) approaches infinity, especially in calculus and analysis. Here, you are looking at how outputs of a function behave as its inputs grow indefinitely.
A sequence might grow, shrink, or oscillate. The limit, if it exists, is the value the sequence will approach as \(n\) turns very large. For instance, if a sequence approaches 5 as \(n\) grows, the limit is 5. In our example, the limit talks about the expression \(\{x\} + \{2x\} + \ldots + \{nx\}\).
It's important to note that for the specific problem given, the outer expression instead takes the form of a quotient, presented as the series over \(n^2\). Thus, we're really caring about how this sum behaves when it's divided by \(n^2\) as \(n\) approaches infinity.
A sequence might grow, shrink, or oscillate. The limit, if it exists, is the value the sequence will approach as \(n\) turns very large. For instance, if a sequence approaches 5 as \(n\) grows, the limit is 5. In our example, the limit talks about the expression \(\{x\} + \{2x\} + \ldots + \{nx\}\).
It's important to note that for the specific problem given, the outer expression instead takes the form of a quotient, presented as the series over \(n^2\). Thus, we're really caring about how this sum behaves when it's divided by \(n^2\) as \(n\) approaches infinity.
Bounded Functions
When dealing with bounded functions, it means the function's values are contained within a fixed range. The precise intuitive meaning is that the function cannot shoot off to infinity or drop to negative infinity.
In the exercise, the fractional part is a perfect example of a bounded function, as each element, e.g., \(\{kx\}\), is between 0 and 1. This bounded nature plays a pivotal role in applying the Squeeze Theorem.
Knowing that your function is bounded helps when analyzing sequences and their behavior since it provides constraints and definitive upper and lower bounds for operations like limits and sums. Indeed, this bounded characteristic is used in the solution to inform the assumption that the fractional terms sum up to at most \(n\), leading to an overall guarantee that the entire sum is beneath another power of n.
In the exercise, the fractional part is a perfect example of a bounded function, as each element, e.g., \(\{kx\}\), is between 0 and 1. This bounded nature plays a pivotal role in applying the Squeeze Theorem.
Knowing that your function is bounded helps when analyzing sequences and their behavior since it provides constraints and definitive upper and lower bounds for operations like limits and sums. Indeed, this bounded characteristic is used in the solution to inform the assumption that the fractional terms sum up to at most \(n\), leading to an overall guarantee that the entire sum is beneath another power of n.
Other exercises in this chapter
Problem 111
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View solution Problem 112
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View solution Problem 116
Let \(f, g\) and \(h\) be real valued functions defined on an interval \(I \subseteq R\) except possibly for some point \(c\) such that $$ \lim _{x \rightarrow
View solution Problem 117
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View solution