To determine the answer to this question, we need to consider the properties of limits, the definition of a function being defined at a point, and the definition of continuity.
Let's go through each statement one by one.
I. lim f(x) exists as x approaches a.
To check the existence of the limit as x approaches a, we need to evaluate the limit of f(x) as x approaches a from both sides.
Let's consider the left-hand limit first:
lim f(x) as x approaches a- = lim [(x^2 - 2ax + a^2) / (x-a)] as x approaches a-
To evaluate this limit, we can simplify the expression by canceling the common factor (x-a) in both the numerator and denominator:
lim f(x) as x approaches a- = lim [(x-a)(x-a) / (x-a)] as x approaches a-
Now, we have:
lim f(x) as x approaches a- = lim (x-a) as x approaches a-
Similarly, let's consider the right-hand limit:
lim f(x) as x approaches a+ = lim [(x^2 - 2ax + a^2) / (x-a)] as x approaches a+
Again, we can simplify the expression:
lim f(x) as x approaches a+ = lim (x-a) as x approaches a+
Since both the left-hand limit and the right-hand limit are equal, we can conclude that the limit of f(x) as x approaches a exists.
II. f(a) exists.
To check if f(a) exists, we simply need to evaluate the function at x = a.
So, let's substitute x=a into the given function:
f(a) = [(a^2 - 2a(a) + a^2) / (a-a)] = [(a^2 - 2a^2 + a^2) / 0]
Here, we encounter a problem. The denominator is 0, which means the function is undefined at x=a. Therefore, f(a) does not exist.
III. f(x) is continuous at x = a.
A function is considered continuous at a point if three conditions hold:
1. The function is defined at that point (f(a) exists).
2. The limit of the function as x approaches that point exists (lim f(x) as x approaches a exists).
3. The value of the limit matches the value of the function at that point (f(a) = lim f(x) as x approaches a).
Since f(a) does not exist, the function fails to meet the condition of being defined at x=a. Therefore, f(x) is not continuous at x = a.
Based on the above analysis, the answer to the question is A. None.