limx趋向正无穷[(1+x)^3/2]/[x^1/2]-x答案是3/2

Question

lim x趋向正无穷 [(1+x)^3/2]/[x^1/2]-x 答案是3/2

Answer 1

lim x->+∞ [(1+x)^3/2]/[x^1/2]-x
=lim x->+∞ [(1+x)^(3/2)-x^(3/2)]/[x^(1/2)] (∞/∞,用罗比达法则)
=lim x->+∞ [3/2*(1+x)^(1/2)-3/2*x^(1/2)]/[1/2*x^(-1/2)]
=lim x->+∞ 3*[(1+x)^(1/2)-x^(1/2)]*x^(1/2)]
=lim x->+∞ 3*[√(1+x)-√x]*√x 分母分子同乘以[√(1+x)+√x]
=lim x->+∞ 3*[(1+x)-x]*√x /[√(1+x)+√x]
=lim x->+∞ 3√x /[√(1+x)+√x]
=lim x->+∞ 3 /[√(1/x+1)+1]
=3/(1+1)=3/2