What is the limit of this function

lim x--> infinity of [sqrt[x^2 + 48) - sqrt(x^2 - 36)]?

does it exist?

If b^2 < a^2 then the binomial expansion of
(a + b)^n = a^n + na^(n - 1)b + [n(n - 1)/2]a^(n - 2)b^2 + ...
Since x ---> infinity, we can assume that x^2 > 48^2 and x^2 > 36^2
sqrt(x^2 + 48) = (x^2 + 48)^(1/2)
= (x^2)^(1/2) + (1/2)(x^2)^(-1/2)*48 + O(x^2)^(-3/2)
= x + 24/x + O(1/x^3)
where O(1/x^3) means 'terms of order 1/x^3 or higher'

Similarly,
sqrt(x^2 - 36) = (x^2 - 36)^(1/2)
= (x^2)^(1/2) + (1/2)(x^2)^(-1/2)*(- 36) + O(x^2)^(-3/2)
= x - 18/x + O(1/x^3)

So
sqrt(x^2 + 48) - sqrt(x^2 - 36)
= [x + 24/x + O(1/x^3)] - [x - 18/x + O(1/x^3)]
= 42/x + O(1/x^3)

lim [sqrt(x^2 + 48) - sqrt(x^2 - 36)] =
x ---> infinity

lim [42/x + O(1/x^3)] = 0
x ---> infinity