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

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Now we see that ( 1- sqrt  x)/ (1- x)

=> (1 – sqrt x)/(1- sqrt x)*(1+ sqrt x)

=> 1/ (1+ sqrt x)

For lim x --> 1 [( 1- sqrt x)/ (1- x)]

we have lim x --> 1 [1/ ( 1+ sqrt x)]

=> 1 / (1 + sqrt 1)

=> 1/ (1+1)

=> 1/2

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

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