Problem 50 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(f(x)=\left\\{\begin{array}{cc}\frac{e^{[x]+|x|}-2}{[x]+|x|}, & x \neq 0 \\\ -1, & x=0\end{array},([.]\right.\) 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

(D) None of these

Step 1: Understand the Problem

We need to determine if the function \( f(x) \) is continuous at \( x = 0 \) or find the limit of \( f(x) \) as \( x \) approaches 0 from the negative or positive side. The function involves the greatest integer function \([x]\) and the absolute value \(|x|\).

Step 2: Evaluate Case When \( x = 0 \)

For \( x = 0 \), the function \( f(x) = -1 \). This is straightforward from the definition given in the problem.

Step 3: Explore \( \\lim_{x \rightarrow 0^{+}} f(x) \)

For \( x \) approaching 0 from the positive side, \([x] = 0\) and \( |x| = x \), so the function becomes \(\frac{e^x - 2}{x}\). We can use L'Hôpital's Rule to find the limit: \[\lim_{x \rightarrow 0^{+}} \frac{e^x - 2}{x} = \lim_{x \rightarrow 0^{+}} \frac{e^x}{1} = e^0 = 1\].

Step 4: Explore \( \\lim_{x \rightarrow 0^{-}} f(x) \)

For \( x \) approaching 0 from the negative side, \([x] = -1\) and \(|x| = -x\), so the function becomes \(\frac{e^{-1-x} - 2}{-1-x}\). This simplifies to \[\lim_{x \rightarrow 0^{-}} \frac{e^{-1-x} - 2}{-1-x} = \frac{e^{-1} - 2}{-1} = 2 - \frac{1}{e}\].

Step 5: Determine Continuity at \( x = 0 \)

Continuity at \( x = 0 \) requires that \( \lim_{x \rightarrow 0} f(x) = f(0) = -1 \). Since we found the limits as \( x \to 0^+ \) and \( x \to 0^- \) are 1 and \(2 - \frac{1}{e}\), respectively, they do not equal \( f(0) \), therefore \( f(x) \) is discontinuous at \( x = 0 \).

Step 6: Identify the Correct Statement

From the previous steps, we see that \( \lim_{x \rightarrow 0^{-}} f(x) = 2 - \frac{1}{e} \), which is not -1 or 1. Thus, none of the provided options correctly describe \( f(x) \) at \( x = 0 \).

Key Concepts

Limit of a function

The concept of the limit of a function is fundamental in calculus. A limit is the value that a function (or sequence) approaches as its input (or index) approaches some value. When you say \lim_{x \to a} f(x) = L, it means that as 'x' gets arbitrarily close to 'a', \( f(x) \) gets arbitrarily close to 'L'.Limits are essential for defining continuity and derivatives. In this exercise, we explored limits from both the positive and negative sides, which are known as one-sided limits.

• A two-sided limit exists if both one-sided limits exist and are equal.
• The two-sided limit of a function at a point is a pivotal factor in determining the function's continuity at that point.

For the function \( f(x) \), we calculated \lim_{x \to 0^+} f(x) = 1 and \lim_{x \to 0^-} f(x) = 2 - \frac{1}{e}. These are not equal, which leads to the conclusion that the overall limit does not exist at \( x = 0 \). This explains why \( f(x) \) is discontinuous at \( x = 0 \), as a function must have an equal two-sided limit at a point to be continuous there.

Greatest integer function

The greatest integer function, denoted by \([x]\), returns the largest integer less than or equal to 'x'. It is sometimes called the floor function. The greatest integer function creates a step-like graph where its value jumps at each integer.Here’s how it works:

• If \( x \) is a positive number, \([x]\) gives the integer part of \( x \). For example, if \( x = 2.9 \), then \([x] = 2\).
• If \( x \) is a negative number, \([x]\) is the first integer that is less than \( x \). For example, if \( x = -1.3 \), then \([x] = -2\).

In this problem, when we looked at \( x \to 0^+ \), we found that \([x] = 0\); and for \( x \to 0^- \), we had \([x] = -1\). These played a crucial role in breaking down the function into separate cases for determining the limits as \( x \to 0 \). The property of taking a sudden leap at integer values makes the greatest integer function special and sometimes complex to handle in continuous and differentiable contexts.

Absolute value function

The absolute value function, denoted as |x|, measures how far a number is from zero on the number line, ignoring its sign. Its value is always nonnegative. The function is defined by:

• \( |x| = x \) if \( x \ge 0 \)
• \( |x| = -x \) if \( x < 0 \)

This piecewise nature allows the absolute value function to transform the input into a purely positive form or zero. It plays a key role in analyzing distances and can introduce non-linear behavior into functions.In the exercise, the absolute value function was crucial when examining \( f(x) \) as \( x \) approached zero. For \( x \rightarrow 0^+ \), we found that |x| = x. For \( x \rightarrow 0^- \), we determined |x| = -x. Each adjusted the value of the function accordingly, impacting the limits from the left and right, and thus the overall continuity evaluation.