$lim_{xto-infty}(sqrt(x^2-3x+1)-sqrt(x^2+x-2))$

A cura di: Administrator

Limite in forma indeterminata $infty – infty$

$lim_{x rightarrow -infty} sqrt{x^2-3x+1}-sqrt{x^2+x-2} =$$=lim_{x rightarrow -infty} frac{x^2-3x+1-x^2-x+2}{sqrt{x^2-3x+1}+sqrt{x^2+x-2}} =$$lim_{x rightarrow -infty} frac{-4x+3}{|x|cdot Big(sqrt{1-frac{3}{x}+frac{1}{x^2}}+sqrt{1+frac{1}{x}-frac{2}{x^2}}Big)} = text{[x < 0]} =$$lim_{x rightarrow -infty} frac{-4x+3}{-xcdot Big(sqrt{1+frac{1}{x}-frac{2}{x^2}}+sqrt{1-frac{3}{x}+frac{1}{x^2}}Big)}=$$=lim_{x rightarrow -infty} frac{-4+frac{3}{x}}{sqrt{1+frac{1}{x}-frac{2}{x^2}}+sqrt{1-frac{3}{x}+frac{1}{x^2}}} = 2$