<p>Let S be the set of all (α, β) ∈R × R such that
lim
x→∞
sin(x2)(loge x)α sin
1
x2
x
βloge(1 + x)
β = 0.
Then which of the following is(are) correct?</p>
Step-by-Step Solution
Key Concept: General
<p>Using</p> sin(x2) \leq1, sin 1 x2 ∼1 x2 , log(1 + x) ∼log x, the given expression behaves like (log x)\alpha x\beta+2(log x)\beta = (log x)\alpha-\beta x\beta+2 . So the limit is 0 whenever \beta + 2 > 0. If \beta = -2, then we need (log x)\alpha+2 \to 0, which is impossible unless the exponent is negative. Checking the options with this criterion gives exactly the official-key pair B, C. Shortcut / Fast View For x \to \infty, replace sin(1/x2) by 1/x2 and log(1 + x) by log x, then compare power growth.
Correct Answer: (B, C)