jee-advanced 2024 Q5

jee-advanced · India · paper2 4 marks Sequences and series, recurrence and convergence Convergence proof and limit determination

Let $S$ be the set of all $( \alpha , \beta ) \in \mathbb { R } \times \mathbb { R }$ such that
$$\lim _ { x \rightarrow \infty } \frac { \sin \left( x ^ { 2 } \right) \left( \log _ { e } x \right) ^ { \alpha } \sin \left( \frac { 1 } { x ^ { 2 } } \right) } { x ^ { \alpha \beta } \left( \log _ { e } ( 1 + x ) \right) ^ { \beta } } = 0 .$$
Then which of the following is (are) correct?
(A) $( - 1,3 ) \in S$
(B) $( - 1,1 ) \in S$
(C) $( 1 , - 1 ) \in S$
(D) $( 1 , - 2 ) \in S$

Let $S$ be the set of all $( \alpha , \beta ) \in \mathbb { R } \times \mathbb { R }$ such that

$$\lim _ { x \rightarrow \infty } \frac { \sin \left( x ^ { 2 } \right) \left( \log _ { e } x \right) ^ { \alpha } \sin \left( \frac { 1 } { x ^ { 2 } } \right) } { x ^ { \alpha \beta } \left( \log _ { e } ( 1 + x ) \right) ^ { \beta } } = 0 .$$

Then which of the following is (are) correct?\\
(A) $( - 1,3 ) \in S$\\
(B) $( - 1,1 ) \in S$\\
(C) $( 1 , - 1 ) \in S$\\
(D) $( 1 , - 2 ) \in S$

Paper Questions

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