Problem 15 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

For each \(\mathrm{t} \in \mathbf{R}\), let \([\mathrm{t}]\) be the greatest integer less than or equal to t. Then, \(\lim _{x \rightarrow 1+} \frac{(1-|x|+\sin |1-x|) \sin \left(\frac{\pi}{2}[1-x]\right)}{|1-x|[1-x]}\) (a) equals 1 (b) equals 0 (c) equals \(-1\) (d) does not exist

Step-by-Step Solution

Answer

The limit equals 1, so the answer is (a).

Step 1: Understand the Expression

The expression involves limits, absolute values, the greatest integer function \([t]\), and sine functions. We are interested in evaluating \(\lim _{x \rightarrow 1+} \frac{(1-|x|+\sin |1-x|) \sin \left(\frac{\pi}{2}[1-x]\right)}{|1-x|[1-x]}\). The path of approach is from the right side of 1, meaning \(x > 1\).

Step 2: Evaluate Sub-Expressions Near 1 from the Right

For \(x \rightarrow 1^+\), we have \(|x| = x\). Therefore, \(|1-x| = x-1\). Now consider \([1-x]\) as \([1-x]\) is the greatest integer that is less than or equal to \(1-x\), which equals 0 when \(x \rightarrow 1^+\).

Step 3: Analyze \([1-x]\) on the Interval

For \(x \rightarrow 1^+\), as \(x\) is slightly greater than 1, \(1-x\) is slightly less than 0, and thus \([1-x] = -1\) for any small positive epsilon added to 1. So, \(\sin\left(\frac{\pi}{2}[1-x]\right) = \sin\left(\frac{\pi}{2} \cdot (-1)\right) = \sin\left(-\frac{\pi}{2}\right) = -1\).

Step 4: Simplify the Fraction

Substitute into the expression with \([1-x] = -1\). Thus, the expression is \(\frac{(1-x+\sin |1-x|)(-1)}{(x-1)(-1)}\), simplifying to \(\frac{1-x+\sin|1-x|}{x-1}\).

Step 5: Evaluate the Limit

As \(x \rightarrow 1^+\), we approximate \(\sin |1-x| \approx |1-x|\) due to small angle approximation, thus becoming negligible in the evaluation. Hence, the main expression simplifies to \(\frac{1-x-x+1}{x-1} = 1\).

Step 6: Conclusion: Determine the Limit Value

The limit simplifies to 1.

Key Concepts

Absolute Value

The absolute value function is a mathematical expression that describes a number's distance from zero on a number line. It is always a non-negative value, irrespective of whether the original number is positive or negative. The absolute value of a number \( x \) is denoted by \( |x| \).

• If \( x \geq 0 \), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \).

When considering limits, absolute values can affect the convergence and continuity of an expression. In the given exercise, as \( x \to 1^+ \), we examine \( |1-x| \). Since \( x \) is slightly greater than 1, we find that \( |1-x| = x - 1 \) because \( 1-x \) becomes negative and absolute value negates the negative sign. Understanding this transformation is essential to simplify expressions and evaluate limits correctly.

Greatest Integer Function

The greatest integer function, sometimes denoted as \( [x] \), refers to the largest integer that is less than or equal to \( x \). This function rounds down any real number to its nearest integer below.

• For example, if \( x = 2.7 \), then \( [x] = 2 \).
• If \( x = -1.3 \), then \( [x] = -2 \).

In our given exercise, evaluating \( [1-x] \) as \( x \to 1^+ \) is crucial. Here, \( 1-x \) is slightly less than 0, making \( [1-x] = -1 \). It is important because the function greatly affects the compound expression within the limit. Understanding when expressions take on integer values can dramatically change the results of operations like sine, which we further dissect.

Sine Function

The sine function, one of the core trigonometric functions, describes a repeating wave that oscillates between -1 and 1. It is a critical component in trigonometry and is used in numerous mathematical contexts, including complex analyses like evaluating limits.The sine of an angle \( \theta \) can be determined using the formula:\[ \sin(\theta) = \text{opposite side over hypotenuse when in a right triangle} \]In this problem, we evaluate \( \sin\left(\frac{\pi}{2}[1-x]\right) \) as \( x \to 1^+ \). Given \([1-x] = -1\), this results in \( \sin\left(-\frac{\pi}{2}\right) = -1 \). Understanding how the greatest integer function influences the angle processed by the sine function is vital to solving the problem's limit. The sine's periodic nature can drastically alter outputs, making it key to carefully track these values.

Limits

Limits are foundational concepts in calculus and analysis that describe the behavior of functions as inputs approach a particular value. They serve as the basis for defining derivatives, integrals, and continuity.A limit can be expressed as:\[ \lim_{x \to a} f(x) = L \]This notation signifies that as \( x \) approaches \( a \), the function \( f(x) \) converges to \( L \).In the exercise, the expression involves limits evaluated as \( x \to 1^+ \). The main task is to simplify the complex expression enclosing an interplay of absolute values, greatest integer functions, and trigonometric computations. By breaking down the inner functions, we derive that the overall limit becomes 1. Grasping how each sub-part influences the larger expression is key to solving such problems accurately.