一道函数题f(x)={[根号(x+4)]-2}/x(x不=0)f(x)=k(x=0)在点x=0处连续,则k=?

Question

一道函数题
f(x)={[根号(x+4)]-2}/x (x不=0)
f(x)=k (x=0)
在点x=0处连续,则k=?

Answer 1

要使f(x)在x=0处连续,则当x趋于0时
lim f(x) = f(0) = k
x趋于0时
lim f(x)
=lim [√(x+4)-2]/x
=lim [√(x+4)-2][√(x+4)+2]/[x(√(x+4)+2)]
=lim [(x+4)-4]/[x(√(x+4)+2)]
=lim 1/[√(x+4)+2]
=1/[√(0+4)+2]
=1/4
所以k = lim f(x) = 1/4