Problem 32 - Calculus for JEE ADVANCED

Question

\(\lim _{x \rightarrow 1} \frac{x \sin (x-[x])}{x-1}\)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understand the Expression

We need to find the limit as \(x\) approaches 1 of \(\frac{x \sin (x - [x])}{x - 1}\), where \([x]\) is the greatest integer function (also known as the floor function). Notice that as \(x\) approaches 1 (from either direction), \(x - [x]\) approaches 0.

Step 2: Evaluate the Numerator Near the Limit

When \(x\) is near 1, \(x - [x]\) is small, thus \(\sin(x - [x])\) is approximately \(x - [x]\). Therefore, we can approximate the numerator by \(x(x - [x])\), simplifying it when \(x\) is very close to one.

Step 3: Substitution and Simplification

Substitute the approximation \(\sin(x - [x]) \approx x - [x]\) into the expression: \[\frac{x(x - [x])}{x - 1}\].Now, since near 1, \(x - [x] = x - 1\), replace \(x - [x]\) with \(x - 1\) to get: \[\frac{x(x - 1)}{x - 1}\].

Step 4: Simplify the Expression

Cancel the common factor \(x - 1\) in the numerator and denominator:\[\frac{x(x - 1)}{x - 1} = x\].This simplification is valid for \(x eq 1\).

Step 5: Evaluate the Limit

Now, we can evaluate the limit as \(x\) approaches 1 of our simplified expression: since we have the expression \(x\) after cancellation, \(\lim_{x \to 1} x = 1.\)

Key Concepts

Greatest Integer Function

The Greatest Integer Function, also known as the floor function, is denoted by \([x]\). This function returns the greatest integer that is less than or equal to \(x\). For example, \([3.7] = 3\) and \([-2.3] = -3\).

It is a step function that jumps at each integer point and is constant between two consecutive integers. This makes it a useful tool in calculus for analyzing discrete changes at integer boundaries.

• When evaluating limits involving the floor function, it is important to consider the behavior as \(x\) approaches an integer from both sides.
• This helps in recognizing how the function behaves just before and after stepping to the next integer.
• Such properties are essential when reducing complex expressions involving this function.

Floor Function

Closely related to the greatest integer function, the term 'floor function' is another name for the operation defined by \([x]\). It 'floors' a number, translating it to the nearest lower integer.

This function is valuable in contexts where you need a whole number result or when dealing with integers in limit problems.

• The floor function can seem counterintuitive at first because it rounds down rather than rounding to the nearest integer.
• Graphically, it's a step function with jumps at each integer value. This graphical representation helps visualize how the function's output remains constant over intervals and only shifts at precise integer points.
• In calculus, using the floor function often helps simplify expressions, especially when handling periodic functions like sine or cosine.

Indeterminate Forms

Indeterminate forms occur in calculus when the limit of a function assumes an undefined algebraic expression such as \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), or \(\infty - \infty\).

These forms do not outright tell us the limit but signal that further analysis is needed.

• We resolve these inefficiencies by simplifying the expression, using algebraic manipulations such as factoring, amplifying, or applying L'Hôpital's Rule.
• L'Hôpital's Rule is particularly useful, allowing us to differentiate the numerator and the denominator separately until the limit no longer appears as an indeterminate form.
• Understanding indeterminate forms is crucial because it prompts us to think creatively about approaching and resolving complex limits.

Trigonometric Limits

Trigonometric limits involve expressions where the functions sine, cosine, or tangent, play a crucial role. These appear frequently in calculus when dealing with periodic or oscillating functions.

The limits often involve small angle approximations, especially when analyzing the behavior of functions as angles approach zero.

• A fundamental approximation used is: \(\sin(x) \approx x\) when \(x\) is small, which is vital for simplifying expressions like \(\sin(x-[x])\).
• These approximations help in understanding the finer behavior of trigonometric functions near specific points.
• Recognizing and applying these approximations is often the key to simplifying expressions and calculating limits accurately.