Question

Lim x>1 x^2-корень из x/корень и x -1

Answer 1

$displaystyle lim_{x to 1} frac{x^2- sqrt{x}}{ sqrt{x-1}}= lim_{x to 1} sqrt{ frac{(x^2- sqrt{x})^2}{x-1}}= lim_{x to 1} sqrt{ frac{(x^2- sqrt{x})^2(x^2+ sqrt{x} )}{(x-1)(x^2+ sqrt{x} )}}= \\= lim_{x to 1} sqrt{ frac{(x^4-2x^2 sqrt{x} +x)(x^2+ sqrt{x} )}{(x-1)(x+ sqrt{x} )}}=\\= lim_{x to 1} sqrt{ frac{x^6-2x^4 sqrt{x} +x^2+x^4 sqrt{x} -2x^3+x sqrt{x} }{(x-1)(x^2+ sqrt{x} )}}=$
$displaystyle = lim_{x to 1} sqrt{ frac{x^6-x^4 sqrt{x} -x^3+x sqrt{x} }{(x-1)(x^2+ sqrt{x} )}}= lim_{x to 1} sqrt{ frac{x^4(x^2- sqrt{x})-x(x^2- sqrt{x} )}{(x-1)(x^2+ sqrt{x} )}}=\\= lim_{x to 1} sqrt{ frac{(x^4-x)(x^2- sqrt{x} )}{(x-1)(x^2+ sqrt{x} )}}= lim_{x to 1} sqrt{ frac{x(x^3-1)(x^2- sqrt{x} )}{(x-1)(x^2+ sqrt{x} )}}=\\= lim_{x to 1} sqrt{ frac{x(x-1)(x^2+x+1)(x^2- sqrt{x} )}{(x-1)(x^2+ sqrt{x} )}} =$
$displaystyle = lim_{x to 1} sqrt{ frac{x(x^2+x+1)(x^2- sqrt{x} )}{(x^2+ sqrt{x} )}}= sqrt{ frac{1*3*0}{2}}=0$

Answer 2

$displaystyle lim_{x to 1} frac{x^2-sqrt{x}}{sqrt{x-1}}= lim_{x to 1} frac{(x^2-sqrt{x})(x^2+sqrt{x})}{sqrt{x-1}(x^2+sqrt{x})}=lim_{x to 1} frac{x^4-x}{sqrt{x-1}(x^2+sqrt{x})}=\\\=lim_{x to 1} frac{x(x^3-1)}{sqrt{x-1}(x^2+sqrt{x})}=lim_{x to 1} frac{x(x-1)(x^2+x+1)}{sqrt{x-1}(x^2+sqrt{x})}=\\\=lim_{x to 1} frac{xsqrt{x-1}(x^2+x+1)}{(x^2+sqrt{x})}=frac{1sqrt{1-1}(1^2+1+1)}{1^2+sqrt{1}}=frac{0}2=0$