设f(x)=根号x,求lim(h->0)[(f(x+h)-f(x))/h]

Question

设f(x)=根号x,求lim(h->0)[ (f(x+h)-f(x)) / h ]

Answer 1

lim(h->0)[ (f(x+h)-f(x)) / h ]
=lim(h->0){[√(x+h)-√x]/h}
=lim(h->0){[(x+h)-x]/[h(√(x+h)+√x)]}
=lim(h->0){1/[√(x+h)+√x]}
=1/(2√x).
注：实际上,此题是导数的定义.