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

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