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

We calculate the limit substituting x by
1

lim [( 1- sqrt x)/ (1- x)] =  (1-1)/(1-1) =
0/0

We notice that we've get an indetermination
case

We'll solve using L'Hospital
rule:

lim [( 1- sqrt x)/ (1- x)] = lim  ( 1- sqrt x)'/ (1-
x)'

lim  ( 1- sqrt x)'/ (1- x)' = lim 
(-1/2sqrtx)/-1

lim [( 1- sqrt x)/ (1- x)] = lim sqrt
x/2

lim [( 1- sqrt x)/ (1- x)] = sqrt
1/2

lim [( 1- sqrt x)/ (1- x)] =
1/2