Problem 50 - A Complete Resource Book in Mathematics for JEE Main

Question

\(f(x)=\left\\{\begin{array}{cl}\frac{e^{[x]+|x|}-2}{[x]+|x|}, & x \neq 0 \\\ -1, & x=0\end{array},([.]\right.\) denotes the greatest integer function), then (A) \(f(x)\) is continuous at \(x=0\) (B) \(\lim _{x \rightarrow 0^{+}} f(x)=-1\) (C) \(\lim _{x \rightarrow 0^{-}} f(x)=1\) (D) None of these

Step-by-Step Solution

Answer

Options (B) and (C) are correct.

Step 1: Definition of Piecewise Function

The function \(f(x)\) is defined as a piecewise function. For \(x eq 0\), \(f(x) = \frac{e^{[x] + |x|} - 2}{[x] + |x|}\), where \([x]\) represents the greatest integer less than or equal to \(x\) and \(|x|\) is the absolute value of \(x\). For \(x = 0\), the function is given as \(f(0) = -1\).

Step 2: Evaluate f(x) for x > 0 as x approaches 0

Since \(x > 0\), \([x] = 0\) and \(|x| = x\). Thus, \(f(x) = \frac{e^{x} - 2}{x}\). As \(x\) approaches 0 from the positive side, this expression approaches \(-1\). Therefore, \(\lim_{x \to 0^+} f(x) = -1\).

Step 3: Evaluate f(x) for x < 0 as x approaches 0

For \(x < 0\), \([x] = -1\) and \(|x| = -x\), so \(f(x) = \frac{e^{-1 + (-x)} - 2}{-1 - x} = \frac{e^{-1-x} - 2}{-1 - x}\). As \(x\) approaches 0 from the negative side, this limit simplifies to \(1\). Hence, \(\lim_{x \to 0^-} f(x) = 1\).

Step 4: Determine continuity at x = 0

For \(f(x)\) to be continuous at \(x = 0\), \(\lim_{x \to 0^+} f(x)\), \(\lim_{x \to 0^-} f(x)\), and \(f(0)\) must all be equal. Since these values are \(-1\), \(1\), and \(-1\) respectively, the function is not continuous at \(x = 0\).

Step 5: Conclusion based on options

(A) \(f(x)\) is not continuous at \(x = 0\) because the right-hand and left-hand limits do not match. Hence, Option A is incorrect. Option B is correct because \(\lim_{x \to 0^+} f(x) = -1\). Option C is correct because \(\lim_{x \to 0^-} f(x) = 1\). Therefore, the correct options are (B) and (C).

Key Concepts

Limits of Functions

When studying the behavior of functions, the concept of limits helps us understand what a function value approaches as the input nears a certain point. Let's consider a point of interest, for example, where a function might change its value rapidly or where the function is not explicitly defined. Using limits, we can determine what happens to the function's value near these critical points.

For the given function, we need to evaluate \(lim_{x \to 0^+} f(x)\) and \(lim_{x \to 0^-} f(x)\). These expressions denote the values the function approaches as \(x\) gets closer to 0 from the positive and negative sides, respectively. Calculating limits can offer insights into the continuity and potential jumps in the function, which helps us understand the function's overall behavior.

In our particular example, the limit from the right, \(\lim_{x \to 0^+} f(x) = -1\), indicates that as \(x\) approaches 0 from the right, the function nears -1. Similarly, the limit from the left, \(\lim_{x \to 0^-} f(x) = 1\), shows a different behavior when approaching from the opposite direction. These mismatching limits suggest there might be a discontinuity at this point.

Greatest Integer Function

The greatest integer function, often represented as \([x]\), is a function that takes a real number \(x\) and rounds it down to the nearest integer. This seemingly simple operation plays a crucial role in the structure of piecewise functions because it helps determine which expressions should be used based on different regions of \(x\).

For a positive number \(x\), the greatest integer function returns the largest whole number less than or equal to \(x\). For instance, \([3.7]\) would equal 3. Similarly, for negative values, \([x]\) gives the next lower integer, so \([-2.4]\) would result in -3.

In the exercise, the greatest integer function manipulates \([x]\) to interact with the absolute value function \(|x|\), creating different expressions based on whether \(x\) is positive or negative. This demonstrates how the greatest integer function can affect the form and limit evaluation of piecewise functions.

Continuity at a Point

Continuity at a specific point \(x = c\) means that the function does not "jump" or "break" at that point, appearing as an unbroken line or curve in a graph. For a function to be continuous at \(x = c\), these conditions must all hold:

• The function value \(f(c)\) exists.
• The limit \(\lim_{x \to c} f(x)\) exists.
• The limit \(\lim_{x \to c} f(x)\) equals the function value \(f(c)\).

Observing the exercise, we see that while \(f(0) = -1\), the limits \(\lim_{x \to 0^+} f(x) = -1\) and \(\lim_{x \to 0^-} f(x) = 1\) are not equal. This mismatch means the function has a discontinuity at \(x = 0\), despite the function having a defined value there.

Identifying points of discontinuity are crucial in understanding the characteristics of piecewise functions, enabling us to predict how the function will behave around specific values.