lim[(x^5+x^4)^(1/5)-(x^5-x^4)^(1/5)].x趋近于正无穷大,

Question

lim[(x^5+x^4)^(1/5)-(x^5-x^4)^(1/5)].x趋近于正无穷大,

Answer 1

∵由泰勒公式展开,得(符号o(x)表示高阶无穷小)
(1+1/x)^(1/5)=1+(1/5)/x-(2/25)/x²+o(1/x²)
(1-1/x)^(1/5)=1-(1/5)/x-(2/25)/x²+o(1/x²)
∴(1+1/x)^(1/5)-(1-1/x)^(1/5)=(2/5)/x+o(1/x²)
故原式=lim(x->+∞){[x(1+1/x)^(1/5)-x(1-1/x)^(1/5)]}
=lim(x->+∞){[x(1+1/x)^(1/5)-x(1-1/x)^(1/5)]}
=lim(x->+∞){x[(1+1/x)^(1/5)-(1-1/x)^(1/5)]}
=lim(x->+∞){x[(2/5)/x+o(1/x²)]}
=lim(x->+∞)[(2/5)+o(1/x)]
=lim(x->+∞)(2/5)+lim(x->+∞)[o(1/x)]
=2/5+0
=2/5