Problem 22 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\lim _{x \rightarrow n}(-1)^{[x]}\), where \([x]\) denotes the greatest integer less than or equal to \(x\), is equal to (A) \((-1)^{n}\) (B) \((-1)^{n-1}\) (C) 0 (D) Does not exist

Step-by-Step Solution

Answer

The limit does not exist.

Step 1: Understand Floor Function

The notation \([x]\) represents the greatest integer less than or equal to \(x\). This means, for any real \(x\), \([x]\) is the integer part of \(x\). For example, \([2.3] = 2\) and \([-1.6] = -2\).

Step 2: Determine Expression Behavior

Consider the expression \((-1)^{[x]}\). The value of \([x]\) shifts as \(x\) approaches \(n\) from the left and right. The value \((-1)^{[x]}\) will alternate between \(1\) and \(-1\) depending on whether \([x]\) is even or odd.

Step 3: Limit Examination

We analyze the behavior of \((-1)^{[x]}\) as \(x\) approaches \(n\) from different sides. As \(x\) approaches \(n\) from the left (just below \(n\)), \([x] = n-1\) resulting in \((-1)^{n-1}\). From the right (just above \(n\)), \([x] = n\) and \((-1)^{[x]} = (-1)^n\).

Step 4: Check Left-Hand and Right-Hand Limits

The left-hand limit as \(x\) approaches \(n\) is \((-1)^{n-1}\) and the right-hand limit is \((-1)^{n}\). Since the left and right limits are different, the limit \(\lim _{x \rightarrow n}(-1)^{[x]}\) does not exist.

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted as \[ [x] \], is a step function that assigns to each real number, the largest integer less than or equal to that number. In simple terms, it "rounds down" any real number to the nearest integer. This function is also known colloquially as the floor function.

For instance:

• If you have a real number like 2.3, the greatest integer function, \[ [2.3] \], will equal 2.
• For a negative number, such as -1.6, the function \[ [-1.6] \] will not equal -1, as rounding down in negatives goes more negative, resulting in -2.

It's important to note that for any integer input, the greatest integer function returns the integer itself. For example, \[ [5] = 5 \]. This function is discontinuous at integer values, which plays an important role in problems involving limits.

Left-Hand Limit

The left-hand limit of a function examines how function values behave as we approach a particular point from the left, or the lower valued side on the number line. It is denoted as \[ ext{lim}_{x \to c^-} f(x) \].

To understand left-hand limits with the greatest integer function, consider the behavior of \[ (-1)^{[x]} \] as \[ x \] approaches an integer \[ n \] from the left. In this case, \[ [x] \] equals \[ n - 1 \] because the function "jumps" to the integer below. Thus, when calculating our example:

• The left-hand limit results in \[ (-1)^{n-1} \].

Understanding the left-hand limit helps to piece together the behavior of the function at the specific point of interest.

Right-Hand Limit

Conversely, the right-hand limit considers how function values behave as the variable approaches a specific point from the right, or the higher valued side. It's denoted as \[ ext{lim}_{x \to c^+} f(x) \].

In problems involving functions like \[ (-1)^{[x]} \], as \[ x \] approaches an integer \[ n \] from values just greater than \[ n \], \[ [x] \] equals exactly \[ n \]. Therefore:

• The right-hand limit at \[ x \rightarrow n \] is \[ (-1)^n \].

This analysis of right-hand limits is crucial for understanding possible discontinuities or jumps.All combined, both left and right-hand limits are necessary to determine if the limit at an integer exists for functions involving the greatest integer function. If the left-hand and right-hand limits don't match, as demonstrated in this case, the overall limit doesn't exist at that point.