What is lim x--> 0 [ (sqrt (1+x) -1)/x]

The limit `lim_(x-> 0) (sqrt (1+x) -1)/x` has to be
determined.

One way of finding the limit is to use
l'Hospital's rule. Another way is to use factorization.

The
factorized form of x^2 - y^2 = (x - y)(x + y)

`(sqrt (1+x)
-1)/x`

= `(sqrt (1+x) -1)/(1 + x -
1)`

= `(sqrt (1+x) -1)/((sqrt(1 + x))^2 -
1^2)`

= `(sqrt (1+x) -1)/((sqrt(1 + x) - 1)(sqrt(1 + x) +
1))`

= `1/(sqrt(1 + x) +
1)`

`lim_(x-> 0) (sqrt (1+x)
-1)/x`

= `lim_(x-> 0) 1/(sqrt (1+x)
+1)`

At x = 0, `1/(sqrt (1+x) +1) = 1/(sqrt 1 + 1) = 1/(1 +
1) = 1/2`

The required limit `lim_(x-> 0) (sqrt
(1+x) -1)/x = 1/2`