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JEE Advanced 2024Paper 2
Exam
JEE Advanced
Year
2024
Paper
Paper 2
Subject
Mathematics
Chapter
Limits

JEE Advanced 2024 · Paper 2 · Mathematics · Limits

Let S be the set of all (α, β) ∈ ℝ × ℝ such that _ x arrow ∞ (x^2)(...

Let SS be the set of all (α,β)∈R×R(\alpha, \beta) \in \mathbb{R} \times \mathbb{R} such that lim⁡x→∞sin⁡(x2)(log⁡ex)αsin⁡(1x2)xαβ(log⁡e(1+x))β=0\lim _{x \rightarrow \infty} \frac{\sin \left(x^2\right)\left(\log _e x\right)^\alpha \sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta}\left(\log _e(1+x)\right)^\beta}=0. Then which of the following is (are) correct?

Options

  1. A(−1,3)∈S(-1,3) \in S
  2. B(−1,1)∈S(-1,1) \in S
  3. C(1,−1)∈S(1,-1) \in S
  4. D(1,−2)∈S(1,-2) \in S

Correct answer

B. (−1,1)∈S(-1,1) \in S

Step-by-step solution

lim⁡x→∞sin⁡x2⋅(log⁡ex)α⋅sin⁡1x2xαβ⋅(log⁡e(1+x))β=0lim⁡x→∞(log⁡ex)α(log⁡e(x+1))β⋅xαβ+2=0\begin{aligned} & \lim _{x \rightarrow \infty} \frac{\sin x^2 \cdot\left(\log _e x\right)^\alpha \cdot \sin \frac{1}{x^2}}{x^{\alpha \beta} \cdot\left(\log _e(1+x)\right)^\beta}=0 \\ & \lim _{x \rightarrow \infty} \frac{\left(\log _e x\right)^\alpha}{\left(\log _e(x+1)\right)^\beta \cdot x^{\alpha \beta+2}}=0\end{aligned}lim⁡x→∞(log⁡exlog⁡e(x+1))β⋅(log⁡ex)α−βxαβ+2=0lim⁡x→∞(log⁡ex)α−βxαβ+2=0 Put log⁡ex=tlim⁡t→∞tα−β(et)αβ+2=0\begin{aligned} & \lim _{x \rightarrow \infty}\left(\frac{\log _e x}{\log _e(x+1)}\right)^\beta \cdot \frac{\left(\log _e x\right)^{\alpha-\beta}}{x^{\alpha \beta+2}}=0 \\ & \lim _{x \rightarrow \infty} \frac{\left(\log _e x\right)^{\alpha-\beta}}{x^{\alpha \beta+2}}=0 \quad \text{ Put } \log _e x=t \\ & \lim _{t \rightarrow \infty} \frac{t^{\alpha-\beta}}{\left(e^t\right)^{\alpha \beta+2}}=0\end{aligned}AsweknowAs we know\lim _{x \rightarrow \infty} \frac{x}{e^x}=0 \alpha \beta+2>0 \Rightarrow \alpha \beta>-2$
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