Problem 13 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

$$ \begin{aligned} &\lim _{x \rightarrow 1} \frac{x \sin (x-[x])}{x-1}, \text { where }[\cdot] \text { denotes the greatest }\\\ &\text { integer function, is equal to } \end{aligned} $$ (A) 1 (B) \(-1\) (C) \(\infty\) (D) Does not exist

Step-by-Step Solution

Answer

(D) Does not exist (due to lack of differentiability at integers).

Step 1: Understand the greatest integer function

The greatest integer function, denoted as \([x]\), gives the largest integer less than or equal to \(x\). For example, \([1.5] = 1\). In the expression \(x - [x]\), this represents the fractional part of \(x\). Because we approach \(x=1\) from both sides, \(x-[x]\) approaches 0.

Step 2: Analyze the behavior near \(x = 1\)

Near \(x = 1\), the expression \(x - [x]\) is either very slightly less than 1 or equal to 0 if we reach exactly 1.Thus, for numbers very close to 1 but less than 1, \(x - [x] = x - 1\). This makes \(\sin(x-[x])\) approximately equal to \(\sin(0) = 0\).

Step 3: Apply the limit using L'Hôpital's Rule

Since we have the form 0/0 when substituting \(x = 1\), we can apply L'Hôpital's Rule. Differentiate the numerator and the denominator:- The derivative of the numerator \(x\sin(x-[x])\) with respect to \(x\) is \(\sin(x-[x]) + x \cos(x-[x]) (1-[\cdot]')\).- The derivative of the denominator \(x-1\) is 1.Evaluate the limit:

Step 4: Evaluate derivatives

However, since the greatest integer function is not differentiable at integer points, directly applying L'Hôpital's Rule without simplification is complex. Instead, conceptualize that the behavior of \(x\sin(x-[x])\) near 1 is essentially 0 times anything approaching zero, leading to 0.

Step 5: Conclusion based on behavior analysis

As \(\lim_{x \to 1} \frac{x\sin(x-[x])}{x-1}\) simplifies directly into evaluating a factor tending towards 0 because the numerator tends to 0. Hence, the expression behaves consistently as being 0 divided by something non-zero. Thus, the limit simplifies to 0.

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted as \([x]\), provides one of the foundational concepts in mathematics, particularly in calculus. It essentially involves a simple operation: rounding down any real number to the nearest integer less than or equal to that number.
For instance:

• For 2.9, the greatest integer less than or equal to 2.9 is 2, so \([2.9] = 2\).
• For a negative number like -1.3, the function results in \([-1.3] = -2\), because -2 is the largest integer not greater than -1.3.

In calculus problems, especially those involving limits, understanding this function is crucial. When dealing with expressions such as \(x - [x]\), it becomes important to note that this part captures the fractional component of \(x\). This fractional part is always between 0 and 1, crucial for analyzing limit behavior as numbers approach integer values. This understanding helps break down complex expressions into smaller, more manageable parts.

L'Hôpital's Rule

L'Hôpital's Rule is a very handy tool in calculus, particularly when you encounter indeterminate forms such as \(\frac{0}{0}\). It provides a way to evaluate limits of quotients by differentiating the numerator and denominator separately.

Here's how it works:

• To apply L'Hôpital's Rule, first confirm that the function you are evaluating results in an indeterminate form such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) when evaluated at a certain point.
• Next, take the derivative of the numerator and the derivative of the denominator.
• Then, compute the limit of these derivatives.

For example, if you're examining \(\lim_{x \to c} \frac{f(x)}{g(x)}\), and both \(f(c) = 0\) and \(g(c) = 0\), you can compute the limit using \(\lim_{x \to c} \frac{f'(x)}{g'(x)}\) instead, provided the latter limit exists.
L'Hôpital's Rule can simplify complex limit problems significantly, making it an essential tool for students.

Fractional Part of a Number

The fractional part of a number is the component that remains after the integer part has been subtracted. For a real number \(x\), the fractional part is \(x - [x]\), where \([x]\) represents the greatest integer less than or equal to \(x\).

Consider some examples:

• For a number like 3.75, the greatest integer is 3. Therefore, the fractional part is \(3.75 - 3 = 0.75\).
• For 1.2, the fractional part is \(1.2 - 1 = 0.2\).

The concept of fractional parts is often useful in scenarios involving periodic functions, like trigonometric functions that reset after their period. Understanding the fractional part helps evaluate how close a number is to the next integer, which can be crucial when analyzing oscillatory behavior or limits near integer boundaries. Thus, recognizing the fractional part provides a deeper understanding of how numbers behave in different mathematical contexts, particularly in calculus.