求(x→+∞)lim((x+5)^0.5-x^0.5)的极限,能写出具体过程么？

Question

求(x→+∞)lim((x+5)^0.5-x^0.5)的极限,
能写出具体过程么？

Answer 1

lim[x→+∞] [√(x+5)-√x],有理化,上下乘以[√(x+5)+√x]化简
=lim[x→+∞] [√(x+5)-√x][√(x+5)+√x]/[√(x+5)+√x]
=lim[x→+∞] (x+5-x)/[√(x+5)+√x]
=lim[x→+∞] 5/[√(x+5)+√x],由于分母趋向正无穷大,∴整个分式趋向零
=0