$lim_{xto 1}((x-sqrt(x^2-3x+3))/(sqrt(10-x)-3))$

A cura di: Administrator

Limite in forma indeterminata $frac{0}{0}$

$lim_{x rightarrow 1} frac{x-sqrt{x^2-3x+3}}{sqrt{10-x}-3} = lim_{x rightarrow 1} frac{(x-sqrt{x^2-3x+3})cdot (sqrt{10-x}+3)}{(sqrt{10-x}-3)cdot (sqrt{10-x}+3)} =$$lim_{x rightarrow 1} frac{(x-sqrt{x^2-3x+3})cdot (sqrt{10-x}+3)}{10-x-9} =$$= lim_{x rightarrow 1} (sqrt{10-x}+3)cdot lim_{x rightarrow 1} frac{x-sqrt{x^2-3x+3}}{1-x} =$$6cdot lim_{x rightarrow 1} frac{(x-sqrt{x^2-3x+3}) (x+sqrt{x^2-3x+3})}{(1-x) (x+sqrt{x^2-3x+3})} =$$= 6cdot lim_{x rightarrow 1} frac{-3 (1-x)}{(1-x) (x+sqrt{x^2-3x+3})} =$$6cdot frac{-3}{2} = -9$