Find limit of x as x aproaches 0 of f(x)=x/[(radical 1+3x)-1]?

Question

Find limit of x as x aproaches 0 of f(x)=x/[(radical 1+3x)-1]?

Answer 1

lim x / [√(1 + 3x) - 1] = (0/0)

x → 0  

multiply f(x) by [√(1 + 3x) + 1]/[√(1 + 3x) + 1] (= 1):

lim {x / [√(1 + 3x) - 1]} {[√(1 + 3x) + 1]/[√(1 + 3x) + 1]} =

x → 0

lim {x [√(1 + 3x) + 1]} / {[√(1 + 3x) - 1][√(1 + 3x) + 1]} =

x → 0

expand the denominator into:

lim {x [√(1 + 3x) + 1]} / {[√(1 + 3x)]² - 1} =

x → 0

lim {x [√(1 + 3x) + 1]} / (1 + 3x - 1) =

x → 0

lim {x [√(1 + 3x) + 1]} / 3x =

x → 0

cancel x, yielding:

lim [√(1 + 3x) + 1] / 3 =

x → 0

lim (√1 + 1) / 3 =  2/3

x → 0

I hope it helps...

Bye!