MHB Problem using big O notation

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The discussion revolves around determining the values of α in the context of big O notation for the function (ln x1)(x2^2 + x2) as ||X|| approaches 0 and infinity, with X defined in R². Participants clarify the notation, specifically that ||X||^α is interpreted as (x1² + x2²)^(α/2). The challenge lies in analyzing the growth of the function relative to ||X|| raised to the power of α. Insights on the behavior of the logarithmic and polynomial components are sought to establish the relationship. The conversation emphasizes the importance of precise notation in understanding the limits and growth rates involved.
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Functions defines on the plane $\mathbb{R}^2$ or open subsets , using $X=(x_1,x_2)\in\mathbb{R}^2$ asthe coordinates
Find all $\alpha \in \mathbb{R}$ such that $(\ln x_1)(x_2^2+x_2)=O(||X||^{\alpha})$ as $||X||\to 0$.
and $|X|| \to \infty$ (note that $x_1>0)$
 
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It might help to be clear on notation. For instance I assume $||X||^\alpha = (x_1^2 + x_2^2)^{\alpha/2}$. In any case this is interesting, anyone got any ideas?
 
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