Problem 86 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\(\lim _{n \rightarrow \infty} \frac{[n x]}{n}\)

Step-by-Step Solution

Answer

The solution to \(\lim _{n \rightarrow \infty} \frac{[n x]}{n}\) is \(x\).

Step 1: Understanding the Problem

We are asked to compute \(\lim _{n \rightarrow \infty} \frac{[n x]}{n}\) where [x] represents the floor function. The floor function, [nx], gives the largest number that is less than or equal to nx. We need to find this limit as \(n \rightarrow \infty\). The first key is to understand that the floor function [nx] can be written as \(nx - \{ nx \}\), where \{ nx \} is the fractional part of nx.

Step 2: Re-write the Limit

Following from the above, we can write the floor function as \(nx - \{ nx \}\). Thus our limit becomes \(\lim _{n \rightarrow \infty} \frac{n x - \{ nx \}}{n}\). This can be split into two separate limits: \(\lim _{n \rightarrow \infty} x - \lim _{n \rightarrow \infty} \frac{\{ nx \}}{n}\). The first limit is a constant (x) divided by infinity, which yields 0, and the second limit is a number between 0 and 1 (the possible range of the fractional part \{ nx \}) divided by infinity, which also yields 0.

Step 3: Solve the Limit

Now we calculate the two limits separately. First, \(\lim _{n \rightarrow \infty} x\), yields x as there is no n in this expression. Secondly, \(\lim _{n \rightarrow \infty} \frac{\{ nx \}}{n}\) yields 0 because as \(n \rightarrow \infty\), a finite number (the fractional part of nx, which is between 0 and 1) divided by a very large number (n) is close to zero.

Step 4: The Final Answer

We add the results of the two limits to get the final answer. The limit \(\lim _{n \rightarrow \infty} \frac{[n x]}{n}\) is equal to \(x + 0 = x\).

Key Concepts

Floor Function

The floor function, denoted by \([\lfloor x \rfloor ]\), is a fundamental concept in mathematics used for approximating real numbers to the nearest lesser or equal integers. When you take the floor of a number, you're essentially rounding it down to the nearest whole number.
For example:

• \([\lfloor 3.7 \rfloor ] = 3\) because it's the largest integer less than or equal to 3.7.
• \([\lfloor -1.3 \rfloor ] = -2\) which is less than -1.3 and is the next integer below it.

This function is especially useful in dividing processes, where it helps to quantify whole units even when results are in fractional forms. In the limit problem provided, the floor function \([nx]\) calculates the greatest integer less than or equal to \([nx]\), where \([n]\) is approaching infinity. The floor function simplifies calculations by removing the "noise" created by the fractional part of numbers.

Fractional Part

The fractional part of a number, represented as \(\{ x \}\), highlights the portion of a number that lies beyond its floor value.
For any real number \( x \), it can be defined as the difference between the number and its floor function: \(\{ x \} = x - [x].\)

• For example, if \( x = 5.75\), the fractional part \(\{ x \} = 5.75 - 5 = 0.75.\)
• Similarly, for a negative number like \( x = -3.4\), the fractional part is \(\{ x \} = -3.4 - (-4) = 0.6.\)

Thus, the fractional part typically retains values between 0 and 1. In the context of limits, such as \( \lim_{n \rightarrow \infty} \frac{\{ nx \}}{n} \), the fractional portion becomes nearly negligible as \( n \) grows infinitely large. Hence, within large calculations, it simplifies to effectively zero, highlighting the significance of fractional parts in fine-tuning whole number calculations.

Infinity

Infinity is an essential concept not only in mathematics but in understanding how limitless values fit into mathematical operations. When we talk about infinity, we refer to a value that is beyond reach, larger than any finite number.
In calculus, we often consider what happens as numbers grow larger and approach infinity. Limits provide a way to describe the behavior of functions as they approach this place "beyond numbers." For example, as we examine \(\lim_{n \rightarrow \infty} \), we look at trends and outcomes of mathematical expressions as \( n \) becomes infinitely large.

• Infinity often symbolizes growth beyond bounds, where sequences or functions might diverge.
• Similarly, dividing by infinity, such as \( \frac{1}{\infty} \), approaches zero, because a finite number repeatedly divided grows minute.

In this problem, infinity allows us to see how the fractional components become irrelevant, leading us to a more straightforward limit outcome, signaling the power of infinity in simplifying complex mathematical expressions.