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I'm trying to solve the following limit, but plotting the functions shows clearly that I'm wrong. I can't understand what's wrong with my reasoning, although I'm pretty sure I'm missing some basilar thing.

$$\lim_{x \to \infty} \sqrt{x}(\sqrt{x-1}-\sqrt{x}) = \lim_{x \to \infty} (\sqrt{x^2-x}-x) = \lim_{x \to \infty} (x\sqrt{1- \frac{1}{x}}-x) = \lim_{x \to \infty} (x - x) = 0 $$

This is how I'm ding it, but plotting the function shows that the limit should be $-\frac{1}{2}$.

Could someone show me what's wrong with what I'm doing?

orsetto
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    $\lim_{x \to \infty} (x\sqrt{1- \frac{1}{x}}-x)\ne \lim_{x \to \infty} (x - x).$ As $h:=\frac1x\to0,$ $\sqrt{1-h}=1-\frac h2+o(\frac1h).$ – Anne Bauval Jul 02 '23 at 09:38
  • Expanse first with $$\frac{\sqrt{x-1} + \sqrt{x}}{\sqrt{x-1} + \sqrt{x}}$$ and then use binomial formular. – Gono Jul 02 '23 at 09:41
  • Or directly: $\sqrt{x}(\sqrt{x-1}-\sqrt{x})=-\frac1{\sqrt{1-\frac1x}+1}$ – Anne Bauval Jul 02 '23 at 09:45

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