Final answer:
By rationalizing the expression for f(x), it is shown that the limit of f(x) as x approaches 0 is -1 / (2 * \sqrt{5}).
Step-by-step explanation:
The function f(x) = (\sqrt{5-x} - \sqrt{5}) / x is not defined at x = 0, but we can find its limit as x approaches 0. To do this, we use the conjugate of the numerator, which is \sqrt{5-x} + \sqrt{5}, to rationalize our expression.
Multiplying both the numerator and the denominator by the conjugate, we get:
(\sqrt{5-x} - \sqrt{5}) / x * (\sqrt{5-x} + \sqrt{5}) / (\sqrt{5-x} + \sqrt{5})
The product of the numerators will be:
(5-x - 5), which simplifies to -x.
Thus the original expression becomes:
-x / [x(\sqrt{5-x} + \sqrt{5})], and this simplifies to:
-1 / (\sqrt{5-x} + \sqrt{5}).
As x approaches 0, \sqrt{5-x} approaches \sqrt{5}, so the expression approaches:
-1 / (2\sqrt{5}) = -1 / (2 * \sqrt{5}).
Consequently, the limit of f(x) as x approaches 0 is -1 / (2 * \sqrt{5}).