lim x approaches 0

Question

[1/x *((1/(sqrt(1+x))-1)]

It will be easier to understand if you write it out..
How would I algebraically do this? (The answer is -1/2)

Help please?

Answer 1

I suggest you use L'Hopital's rule. The limit of the ratio of (1/(sqrt(1+x))-1)] to x is the ratio of the derivatives of numerator and denominator.
The derivative of x (the denominator) is 1.
The derivative of 1/(sqrt(1+x) -1 at x = 0 is (-1/2)(1+x)^-3/2 = -1/2
So the limit is (-1/2)/1 = -1/2