Problem 49 - A Complete Resource Book in Mathematics for JEE Main

Question

The function \(f(x)=\left\\{\begin{array}{l}(x+1)^{2-\left(\frac{1}{|x|}+\frac{1}{x}\right)}, x \neq 0 \text { is } \\ 0 & , x=0\end{array}\right.\), (A) discontinuous at only one point (B) discontinuous exactly at two points (C) continuous everywhere (D) None of these

Step-by-Step Solution

Answer

The function is discontinuous only at \( x=0 \).

Step 1: Understand the Function

The function \( f(x) \) is defined piecewise. For \( x eq 0 \), it is given by \( (x+1)^{2 - \left( \frac{1}{|x|} + \frac{1}{x} \right)} \). At \( x = 0 \), the function is simply defined as 0.

Step 2: Analyze the Function for x ≠ 0

For \( x eq 0 \), we need to consider the expression \( \frac{1}{|x|} + \frac{1}{x} \). This expression is undefined at \( x = 0 \) and problematic for very small values of \( x \).

Step 3: Check Continuity at x = 0

When \( x = 0 \), the function \( f(x) = 0 \). We must check if the limit as \( x \to 0 \) is equal to \( f(0) = 0 \).

Step 4: Find the Limit as x Approaches 0

Calculate the one-sided limits:- As \( x \to 0^+ \), \( \frac{1}{|x|} + \frac{1}{x} \to +\infty \).- As \( x \to 0^- \), \( \frac{1}{|x|} + \frac{1}{x} \to -\infty \).For these cases, the base, \( x+1 \), experiences exponents approaching \(-\infty\) and \(+\infty\), making the behavior vary between 0 and \( \infty \). The limits do not match.

Step 5: Determine Continuity

Since the left-hand limit does not equal the right-hand limit, \( f(x) \) is not continuous at \( x = 0 \). Therefore, \( x = 0 \) is a point of discontinuity.

Step 6: Check Continuity at Other Points

For all other \( x eq 0 \), the expression \( 2 - \left( \frac{1}{|x|} + \frac{1}{x} \right) \) does not introduce discontinuities, provided the base \( x+1 \) is positive. As long as \( x eq -1 \), hence, the function is also defined for nonzero points.

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions defined by different formulas over different intervals. This means that the rule for calculating the function's value changes depending on the value of the input. In our example, the function \( f(x) \) is defined piecewise with two separate expressions:

• For \( x eq 0 \), the expression is \((x+1)^{2 - \left( \frac{1}{|x|} + \frac{1}{x} \right)}\).
• For \( x = 0 \), the function is simply 0.

This type of definition allows the function to be flexible, adapting to different scenarios. It can accommodate areas where a single formula might not work well across a full range of values. Understanding how the pieces of a piecewise function fit together is crucial for analyzing characteristics like continuity and limits.

Limits at a Point

When discussing continuity, understanding limits is fundamental. The limit of a function as it approaches a particular point allows us to predict the behavior of the function at that point. For piecewise functions, evaluating limits at transition points, such as \( x = 0 \) in our case, helps determine if the function is continuous. For \( f(x) \) as \( x \to 0 \), we evaluate the one-sided limits:

• As \( x \to 0^+ \), the expression \( \frac{1}{|x|} + \frac{1}{x} \) tends towards \( +\infty \).
• As \( x \to 0^- \), it tends towards \( -\infty \).

These differing limits imply that the function values near \( x = 0 \) can be quite varied. A consistent, well-defined limit at a particular point supports continuity at that point, which is not the case here as the side limits do not match.

Discontinuity Analysis

Discontinuity in a function occurs when there is a break, gap, or jump at a point. To find if a function is continuous at a point, you check if the three following criteria are met:

• The function is defined at the point.
• The limit as it approaches the point exists.
• The limit value equals the function value at that point.

In our example at \( x = 0 \), \( f(x) \) is not continuous because the criteria fail:- The limit from the positive side does not equal the limit from the negative side.- Thus, \( x = 0 \) is a point of discontinuity because the function does not meet the requirements for continuity at this junction.

Mathematical Functions

Mathematical functions are relationships between inputs (often real numbers) and outputs (also real numbers). They provide the framework for analyzing mathematical behavior across different variables. In the presented problem, \( f(x) \) operates as a mathematical function defined with a piecewise rule. The study of such functions involves exploring their domains and ranges, as well as continuity and differentiability. Mathematical functions can be generalized, but specific characteristics like those in piecewise functions need careful investigation, especially when they toggle formulae around certain points (like \( x = 0 \) in our example). By understanding these functions, we can predict their behavior and solve various problems in calculus and algebra.