Limits
Limit of product with powers — pairs (α,β) in set S
MJAT_TS7_P2
Grade 12
Question:
Let $S$ be the set of all $(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ such that $\displaystyle\lim_{x\to\infty}\frac{\sin(x^\alpha\beta^2)(\log_e(x^{\alpha}+\sin x))}{(\beta x^{1/20})} = 0$. Then which of the following is/are correct?
A) $(-1,3)\in S$
B) $(-1,1)\in S$
C) $(1,-1)\in S$
D) $(1,-2)\in S$
Step-by-Step Solution
Key Concept: For the limit to be 0 as $x\to\infty$: the numerator $\sin(x^\alpha\beta^2)\cdot\ln(x^\alpha+\sin x)$ divided by $\beta x^{1/20}$ must $\to 0$. If $\alpha<0$: $x^\alpha\to 0$, $\sin(x^\alpha\beta^2)\to 0$, $\ln(x^\alpha+\sin x)\to\ln(\sin x)$ bounded... denominator $\to 0$ if $\beta\neq 0$. Analysis needed for each case.
Answer: B, C.
Correct Answer: BC