Class JEE Mathematics ALL Q #1181
COMPETENCY BASED
APPLY
4 Marks 2026 JEE Main 2026 (Online) 21st January Morning Shift MCQ SINGLE
If $f(3)=18$, $f'(3)=0$ and $f''(3)=4$ then the value of $\lim_{x\rightarrow1}\ln(\frac{f(x+2)}{f(3)})^{\frac{18}{(x-1)^{2}}}$ is:
(A) $2$
(B) $4$
(C) $6$
(D) $8$
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AI Tutor Explanation

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Step-by-Step Solution

  1. Let $L = \lim_{x\rightarrow1}\ln(\frac{f(x+2)}{f(3)})^{\frac{18}{(x-1)^{2}}}$.

  2. Then $L = \lim_{x\rightarrow1} \frac{18}{(x-1)^2} \ln(\frac{f(x+2)}{f(3)})$.

  3. Let $x-1 = h$, so $x = h+1$. As $x \rightarrow 1$, $h \rightarrow 0$.

  4. Then $L = \lim_{h\rightarrow0} \frac{18}{h^2} \ln(\frac{f(h+3)}{f(3)})$.

  5. Using Taylor series expansion for $f(h+3)$ around $h=0$, we have $f(h+3) = f(3) + hf'(3) + \frac{h^2}{2}f''(3) + O(h^3)$.

  6. Given $f(3) = 18$, $f'(3) = 0$, and $f''(3) = 4$, we have $f(h+3) = 18 + 0 + \frac{h^2}{2}(4) + O(h^3) = 18 + 2h^2 + O(h^3)$.

  7. So, $L = \lim_{h\rightarrow0} \frac{18}{h^2} \ln(\frac{18 + 2h^2 + O(h^3)}{18}) = \lim_{h\rightarrow0} \frac{18}{h^2} \ln(1 + \frac{2h^2}{18} + O(h^3))$.

  8. Using the approximation $\ln(1+x) \approx x$ for small $x$, we have $L = \lim_{h\rightarrow0} \frac{18}{h^2} (\frac{2h^2}{18} + O(h^3))$.

  9. Therefore, $L = \lim_{h\rightarrow0} \frac{18}{h^2} \cdot \frac{2h^2}{18} = 2$.

Correct Answer: 2

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AI Suggestion: Option A

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply L'Hopital's rule and Taylor series expansion to evaluate the limit, given the function's values and derivatives at a specific point.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to execute a sequence of steps, including applying L'Hopital's rule and using Taylor series expansion, to arrive at the solution. It's about knowing how to do something.
Syllabus Audit: In the context of JEE, this is classified as COMPETENCY. It requires application of concepts like limits, derivatives, and Taylor series to solve a non-standard problem, testing problem-solving skills rather than direct recall of formulas.

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