<s>[INST] How to calculate limit? I'm puzzled with this limit. The answer is -0.5, but how to get it? $\lim_\limits{x \to \infty}1-x+\sqrt{\frac{x^3}{x+3}}$
 [/INST] I've tried to multiply by conjugate not only part with a variable, but all the expression, and this way I calculate it without replace and L'Hospital's Rule.
$\lim_\limits{x \to \infty} 1-x+\sqrt{\frac{x^3}{x+3}}=\lim_\limits{x \to \infty}\frac{\frac{x^3}{x+3}-(x-1)^2}{\sqrt{\frac{x^3}{x+3}}+x-1} = \lim_\limits{x \to \infty}\frac{-x^2+5x-3}{(x+3)\left(\frac{\sqrt{x^3}+\sqrt{x+3}(x-1)}{\sqrt{x+3}}\right)}=\lim_\limits{x \to \infty}\frac{-x^2+5x-3}{\sqrt{x^4+3x^3}+(x+3)(x+1)}$
Now we can see easily that factor before $x^2$ is $-1$ in numerator and $2$ in denominator, so the result is $-\frac{1}{2}$
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