What is the limit of  (a^2-x^2)/(sqrt(x)-sqrt(a)) for x tending to a.  

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The value of lim x-->a [ (a^2 – x^2)/(sqrt x – sqrt a)] has to be determined.

Subsituting x = a, gives us 0/0 which cannot be determined.

Notice that x^2 – a^2 = (x – a)(x + a) = (sqrt x – sqrt a)(sqrt x + sqrt a)(x + a)

lim x-->a [ (a^2 – x^2)/(sqrt x – sqrt a)]

=> lim x-->a [-(sqrt x – sqrt a)(sqrt x + sqrt a)(x + a)/(sqrt x – sqrt a)]

=> lim x-->a [-(sqrt x + sqrt a)(x + a)]

substitute x = a

=> -(sqrt a + sqrt a)(a + a)

=> -2*sqrt a*2*a

=> -4*a*sqrt a

The limit we have to determine is equal to -4*a*sqrt a

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