Problem 6 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(f(x)=[x]-\left[\frac{x}{4}\right], x \in \mathrm{R}\), where \([x]\) denotes the greatest integer function, then: \(\quad\) (a) \(f\) is continuous at \(x=4\). (b) \(\lim _{x \rightarrow 4+} f(x)\) exists but \(\lim _{x \rightarrow 4} f(x)\) does not exist. (c) Both \(\lim _{x \rightarrow 4-} f(x)\) and \(\lim _{x \rightarrow 4} f(x)\) exist but are not equal. (d) \(\lim _{x \rightarrow 4-} f(x)\) exists but \(\lim _{x \rightarrow 4+} f(x)\) does not exist.

Step-by-Step Solution

Answer

(a) \(f\) is continuous at \(x=4\).

Step 1: Understanding the Floor Function

The function is given as \( f(x) = [x] - \left[ \frac{x}{4} \right] \), where \([x]\) represents the greatest integer less than or equal to \(x\). This means \([x]\) is the largest integer \(\leq x\), and \(\left[ \frac{x}{4} \right]\) applies the same logic to \( \frac{x}{4} \).

Step 2: Analyzing Limits from the Left

To find \( \lim_{x \to 4^-} f(x) \), evaluate for values approaching 4 from the left, such as 3.9. Here, \([3.9] = 3\) and \(\left[ \frac{3.9}{4} \right] = 0\), thus \( f(3.9) = 3 - 0 = 3 \). For any \( x \) between 3 and 4 (not including 4), \( [x] = 3 \) and \([ \frac{x}{4} ] = 0\), so it remains 3. Thus, \( \lim_{x \to 4^-} f(x) = 3 \).

Step 3: Analyzing Limits from the Right

To find \( \lim_{x \to 4^+} f(x) \), consider values just greater than 4, such as 4.1. Here, \([4.1] = 4\) and \(\left[ \frac{4.1}{4} \right] = 1\), thus \( f(4.1) = 4 - 1 = 3 \). For any \( x \) between 4 and 5, the values for \([x]\) and \(\left[ \frac{x}{4} \right]\) will also give 3 as output. Thus, \( \lim_{x \to 4^+} f(x) = 3 \).

Step 4: Evaluating Two-Sided Limits and Continuity

Calculate \( f(4) = [4] - \left[ \frac{4}{4} \right] = 4 - 1 = 3 \). Since \( \lim_{x \to 4^-} f(x) = 3 \), \( \lim_{x \to 4^+} f(x) = 3 \), and \( f(4) = 3 \), it follows that all limits exist and are equal. Hence, \( f(x) \) is continuous at \( x = 4 \).

Key Concepts

Greatest Integer Function

In mathematics, the greatest integer function, denoted by \([x]\), is a way to map any real number \(x\) to the largest integer less than or equal to \(x\). This function is also known as the floor function. Here are some key points to understand it better:

• For a positive number like 3.76, \([3.76] = 3\) because 3 is the largest integer less than 3.76.
• For a negative number like -2.34, \([-2.34] = -3\) since -3 is the largest integer that is less than -2.34.
• When dealing with whole numbers, \([3] = 3\), since there's no fraction part.

In our given function \(f(x) = [x] - \left[ \frac{x}{4} \right]\), both the input \(x\) and the division result \(\frac{x}{4}\) are mapped by the greatest integer function to the nearest lower or equal integer.

One-Sided Limits

When we talk about one-sided limits in calculus, we're studying the behavior of a function as it approaches a particular point from one side. For a function \(f(x)\), the one-sided limits at a point \(c\) are expressed as:

• Left-hand limit, \( \lim_{x \to c^-} f(x)\), means the limit of \(f(x)\) as \(x\) approaches \(c\) from values smaller than \(c\).
• Right-hand limit, \( \lim_{x \to c^+} f(x)\), considers \(x\) approaching \(c\) from values larger than \(c\).

For \(f(x) = [x] - \left[ \frac{x}{4} \right]\), when evaluating around \(x = 4\), the left-hand limit is computed using values less than 4, while the right-hand limit uses values greater than 4. Both limits appear to yield the same result for this particular function, which is 3.

Two-Sided Limits

The concept of a two-sided limit is about determining what value a function \(f(x)\) approaches as \(x\) nears a specific point from both sides. To say that the limit exists, both the left-hand and right-hand limits must not only exist but also be equal. For any point \(c\), this is expressed as:\[\lim_{x \to c} f(x) = L\]This means as \(x\) nears \(c\) from both directions, the outputs of \(f(x)\) converge to the value \(L\).
In our case, at \(x = 4\), since both one-sided limits were found to be 3, the two-sided limit exists and is equal to 3.
This effectively makes our function continuous at \(x = 4\), which leads us to the concept of continuity.

Limit Existence

The existence of a limit is crucial to understanding the continuity and overall behavior of a function. A limit \( \lim_{x \to c} f(x)\) exists when both of these conditions are met:

• Both the left-hand \( \lim_{x \to c^-} f(x)\) and right-hand \( \lim_{x \to c^+} f(x)\) limits exist.
• The two limits are equal: \( \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)\).

When these requirements are satisfied, we can say \( f(x) \) approaches a single definitive value as \( x \) nears \( c \).
For our function \( f(x) = [x] - \left[ \frac{x}{4} \right] \) at \( x = 4 \), both the left and right limits have been calculated as 3, establishing the existence of the limit at that point. This is a significant detail confirming the continuity of the function at \( x = 4 \).