Question

Let S be the set of all $(α, β) ∈ \R\times\R$ such that
$\lim\limits_{x\rightarrow\infin}\frac{\sin(x^2)(\log_ex)^\alpha\sin(\frac{1}{x^2})}{x^{αβ}(\log_e(1+x))^β}=0$
Then which of the following is (are) correct ?

• A. (-1, 3) ∈ S
• B. (-1, 1) ∈ S
• C. (1, -1) ∈ S
• D. (1, -2) ∈ S

Answer 1

Answer: (B), (C)

(-1, 1) ∈ S

Answer 2

The correct option is (B): (-1, 1) ∈ S and (C): (1, -1) ∈ S.