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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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

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