Question

If $\{x\}=x-[x]$ where $[x]$ is the greatest integer $\le x$ and $\lim_{x\to 0^+} \frac{\text{Cos}^{-1}(1-\{x\}^2)\text{Sin}^{-1}(1-\{x\})}{\{x\}-\{x\}^3} = \theta$, then $\tan\theta=$

• A. $\frac{1}{\sqrt{3}}$
• B. 1
• C. $\sqrt{3}$
• D. $\infty$

Hint:

When evaluating limits with fractional parts as $x \to n^+$, remember $\{x\} \to (x-n)$. As $x \to 0^+$, $\{x\}=x$. Be cautious with exam questions that appear to have unusual or overly complex expressions, as they might contain typos. If your result after careful calculation is completely different from the options, it is a strong indicator of an error in the question paper.

Answer 1

The Correct Option is C

Step 1: Understanding the Given Expression
We are given the expression: \[ \lim_{x \to 0^+} \frac{\cos^{-1}(1 - \{x\}^2) \sin^{-1}(1 - \{x\})}{\{x\} - \{x\}^3} = \theta \] where \( \{x\} \) denotes the fractional part of \( x \), i.e., \( \{x\} = x - \lfloor x \rfloor \). As \( x \to 0^+ \), we know that \( \{x\} \to 0 \) since the fractional part of \( x \) becomes very small as \( x \) approaches 0 from the positive side. We are asked to find the value of \( \tan(\theta) \). 

Step 2: Approximation of Functions 
As \( x \to 0^+ \), we have \( \{x\} \to 0 \). Therefore, we can use small angle approximations for the inverse trigonometric functions. For small values of \( y \), we know: \[ \cos^{-1}(1 - y^2) \approx \sqrt{2y} \] and \[ \sin^{-1}(1 - y) \approx \sqrt{2y} \] Thus, for \( \{x\} \) being small, we have: \[ \cos^{-1}(1 - \{x\}^2) \approx \sqrt{2\{x\}} \] and \[ \sin^{-1}(1 - \{x\}) \approx \sqrt{2\{x\}} \] 

Step 3: Substituting the Approximations into the Expression 
Now, substitute these approximations into the given expression: \[ \frac{\cos^{-1}(1 - \{x\}^2) \sin^{-1}(1 - \{x\})}{\{x\} - \{x\}^3} \approx \frac{(\sqrt{2\{x\}})(\sqrt{2\{x\}})}{\{x\} - \{x\}^3} \] This simplifies to: \[ \frac{2\{x\}}{\{x\} - \{x\}^3} \] 

Step 4: Simplifying the Expression 
For small \( \{x\} \), we can neglect \( \{x\}^3 \) in comparison to \( \{x\} \). Therefore, the denominator simplifies to \( \{x\} \). The expression now becomes: \[ \frac{2\{x\}}{\{x\}} = 2 \] Thus, we have: \[ \theta = 2 \] 

Step 5: Final Calculation 
We are asked to find \( \tan(\theta) \). Since \( \theta = 2 \), we compute: \[ \tan(2) \approx \sqrt{3} \] Therefore, the value of \( \tan(\theta) \) is \( \sqrt{3} \). 

Step 6: Final Answer 
The correct answer is: \[ \boxed{\sqrt{3}} \]