I have an expression

$x^2[log(1+\sqrt x+x)+C], x\in \mathbb{R},>0$

Where C is an arbitrary constant

I want to find whether this expression is asymptotic to $x$ when $x\rightarrow\infty$

So far I tried log inequality properties and taking advantage of choosing C and so on...

But no luck.

Any suggestions or help will be appreciated!

Thank you so much for reading and having interest in my question.

  • $\begingroup$ What do you mean by asymptotic to $x$ when $x$ is large? There are multiple definitions of asymptotic. Please be more specific. $\endgroup$ – Jonathan Davidson Aug 5 '17 at 6:22
  • $\begingroup$ No, certainly not, since the expression is lower bounded by $Cx^2$, which isn't even in $O(x)$. $\endgroup$ – MathematicsStudent1122 Aug 5 '17 at 6:24
  • $\begingroup$ whether $x^2[log(1+\sqrt x+x)+C] \sim x$ as $x\rightarrow\infty, \;\;Arg(x^2[log(1+\sqrt x+x)+C])=0$ or not. Sorry, my math skill cannot express more specific than that. All reals in this question. No imaginary. $\endgroup$ – Duke Smith Aug 5 '17 at 6:25

No, it is not asymptotic to $x$

$$\lim_{x \to \infty} \frac{x^2(\log(1 + \sqrt{x} + x) + C)}{x} = \infty$$


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