Problem 115 - A Complete Resource Book in Mathematics for JEE Main

Question

In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: The function \(f(x)=\) \(\lim _{n \rightarrow \infty} \frac{\cos \pi x-x^{2 n} \sin (x-1)}{1+x^{2 n+1}-x^{2 n}}\) is discontinuous at \(x=\pm 1\) Reason: \(f(x)=\left\\{\begin{array}{cl}\frac{\cos \pi x}{1+x}, & |x|<1 \\\ -1+\sin 2, & x=-1 \\ -1, & x=1 \\ \frac{-\sin (x-1)}{x-1}, & |x|>1\end{array}\right.\)

Step-by-Step Solution

Answer

B: Both Assertion and Reason are true, but Reason is not a correct explanation.

Step 1: Understanding the Assertion

The assertion claims that the function \( f(x) = \lim_{n \rightarrow \infty} \frac{\cos \pi x-x^{2n} \sin (x-1)}{1+x^{2n+1}-x^{2n}} \) is discontinuous at \( x = \pm 1 \). To verify this, we need to compute the limit of the function as \( n \to \infty \) and evaluate the continuity at these points.

Step 2: Analyzing the Function at \( x = -1 \)

Substitute \( x = -1 \) in the function: \[ f(-1) = \lim_{n \to \infty} \frac{\cos(\pi \cdot -1)-(-1)^{2n} \sin(-1-1)}{1+ (-1)^{2n+1}- (-1)^{2n}} \]This simplifies to \[ f(-1) = \lim_{n \to \infty} \frac{-1-\sin(-2)}{1+(-1)^{2n+1}-1} \ = \lim_{n \to \infty} \frac{-1+\sin(2)}{2}\ = -1 + \sin 2 \] Checking reason, when \( x = -1 \), it matches the given output in the reason.

Step 3: Analyzing the Function at \( x = 1 \)

Substitute \( x = 1 \) in the function:\[ f(1) = \lim_{n \to \infty} \frac{\cos(\pi \cdot 1)-1^{2n} \sin(1-1)}{1 + 1^{2n+1} - 1^{2n}} \ = \lim_{n \to \infty} \frac{-1}{1+1-1} \ = -1 \]This results in \( f(1) = -1 \), matching the form given for \( x = 1 \) in the reason.

Step 4: Exploring Continuity Around \( x = \pm 1 \)

The function should return the same value whether approached from the left or right. When substituting values slightly larger and slightly smaller than -1 and 1 respectively, the expected limits should resolve to the defined piece in the reason. Since they correspond correctly, the limits differ thus function may be discontinuous. This matches the assertion's claim.

Step 5: Evaluating Reason Given

The reason states a piecewise function explaining \( f(x) \). The calculated values for \( f(-1) \) and \( f(1) \) match those provided by the reason. This confirms that the reason correctly describes function \( f(x) \) for \( x=-1 \) and \( x=1 \), yet it doesn't explain why \( f(x) \) is discontinuous at these points.

Key Concepts

Assertion and Reason

The concept of assertion and reason revolves around evaluating the truthfulness of propositions and their justifications. In mathematical exercises, it is common to be presented with a statement, known as the assertion, and a linked justification termed as the reason.

When tackling assertion and reason questions, there are common steps to follow:

• First, evaluate the assertion on its own to see if it is true or false. This involves understanding the given mathematical functions or expressions involved.
• Next, assess the reason provided to verify its truth independently of the assertion.
• Finally, determine if the reason actually explains the assertion. A correct reasoning does not always mean it explains the assertion.

In the problem at hand, understanding and evaluating limits and continuity around specific points helps determine the correctness of both the assertion and the reason provided.

In the example given, both the assertion and the reason were found to be true, but the reason didn’t explain why the function was discontinuous at the specified points, hence illustrating the need to deeply comprehend both elements.

Limits and Continuity

The concepts of limits and continuity are fundamental in calculus and analysis. A limit examines the behavior of a function as it approaches a specific point, while continuity ensures no abrupt changes in function value at that point.

To check the continuity of a function at a particular point, we follow these steps:

• First, compute the function's limit as the variable approaches the specific point from both directions (left and right).
• If the limits from both directions equal the actual function value at that point, the function is continuous there.
• If the limits differ, or if the function value isn't equal to the computed limit, the function is discontinuous at that point.

In the given problem, the evaluation involved limits as approaches \(x = \pm 1\). By substituting different values and evaluating respective limits, we checked for continuity at these points. Although the function values and limits were computed correctly, they differed when approached from different directions, indicating discontinuity.

This accentuates the importance of continuity in understanding function behavior over an interval or at specific points.

Piecewise Functions

Piecewise functions are mathematical expressions where different formulas apply to different intervals or regions within the domain. They are essential when functions behave differently over different sections of their input range.

For piecewise functions, it's critical to ensure the function properly transitions between various pieces without jumps or breaks. Here's how we deal with them:

• Identify all distinct pieces and the intervals on which they apply.
• Determine the function value for any boundary points by substituting these points into the respective piece.
• Check the continuity at boundary points by evaluating limits from both sides to ensure they match the function's value at those points.

In the problem we studied, the piecewise function was given to define different behaviors for \(x < -1\), \(-1 \, \leq \, x \, \leq \, 1\), and \(x > 1\). Understanding the behavior of each piece and appropriately checking for continuity is vital to grasping the function's overall character.

Piecewise definitions underscore the flexible nature of mathematical functions and highlight the importance of detailed function analysis in complex scenarios.