Let S be the set of all such that lim x sin ( x 2 ) ( log e x ) sin ( 1 x 2 ) x ( log e ( 1 + x ) ) = 0 . Then which of the following is (are) correct? [2024]

Question

Let $S$ be the set of all $(α, β) \in ℝ \times ℝ$ such that $\lim_{x \to \infty} \frac{\sin (x^{2}) {(\log_{e} x)}^{α} \sin (\frac{1}{x^{2}})}{x^{α β} (\log_{e} {(1 + x))}^{β}} = 0 .$

Then which of the following is (are) correct? [2024]

Smart Achievers · Accepted Answer

(2, 3)

$Given, \lim_{x \to \infty} \frac{\sin (x^{2}) \sin (\frac{1}{x^{2}}) {(\ln x)}^{α}}{x^{α β} (\ln {(1 + x))}^{β}} = 0$

$= \lim_{x \to \infty} \frac{(\sin x^{2}) \sin (\frac{1}{x^{2}}) \frac{1}{x^{2}} {(\ln x)}^{α}}{(\frac{1}{x^{2}}) x^{α β} (\ln {(1 + x))}^{β}} = 0$

$= \lim_{x \to \infty} {(\frac{\ln x}{\ln (1 + x)})}^{β} \cdot \frac{{(\ln x)}^{α - β}}{x^{α β + 2}} = 0$

$= \lim_{x \to \infty} \frac{{(\ln x)}^{α - β}}{x^{α β + 2}} = 0$

$It is possible if α β + 2 > 0 α β > - 2$

Solution

Q.
Let $S$ be the set of all $(α, β) \in ℝ \times ℝ$ such that $\lim_{x \to \infty} \frac{\sin (x^{2}) {(\log_{e} x)}^{α} \sin (\frac{1}{x^{2}})}{x^{α β} (\log_{e} {(1 + x))}^{β}} = 0 .$

Then which of the following is (are) correct [2024]

1 $(- 1, 3) \in S$

2 $(- 1, 1) \in S$

3 $(1, - 1) \in S$

4 $(1, - 2) \in S$

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Solution

Q. Let S be the set of all (α,β)∈ℝ×ℝ such that limx→∞sin(x2)(logex)αsin(1x2)xαβ(loge(1+x))β=0. Then which of the following is (are) correct [2024]

1 (-1,3)∈S

2 (-1,1)∈S

3 (1,-1)∈S

4 (1,-2)∈S