Let f be the function defined as:

f(x) = (x^2 - 4)/ (x-2) if x≠2
1 if x=2
Which of the following statement(s) about f is true?
A. f has limit at x=2
B. f is continuous at x=2
C. f is differentiable at x=2

f(x) = x+2 for all x ≠ 2

so
A is true
B is false
C - what does it mean to be differentiable?

Well, let's take a closer look at this function f(x) = (x^2 - 4) / (x-2), shall we?

First, let's consider statement A: f has a limit at x=2. Well, using algebraic manipulation, we can rewrite f(x) as f(x) = x + 2. However, when we plug in x=2 into the original function, we get an indeterminate form of 0/0. This means that the function does NOT have a limit at x=2. So statement A is false.

Now, let's move on to statement B: f is continuous at x=2. Continuity requires that the function has a limit at x=2 AND that the value of the function equals the limit. Since the limit is not defined at x=2, f cannot be continuous at x=2. So statement B is also false.

Lastly, let's consider statement C: f is differentiable at x=2. Since differentiability requires continuity, and f is NOT continuous at x=2 (as we determined earlier), it follows that f is also not differentiable at x=2. So, statement C is false as well.

To sum it up, none of the given statements A, B, or C are true for the function f in question. It seems f(x) is quite the troublemaker!

To determine which statement(s) about the function f are true, we need to evaluate the properties of the function at x = 2.

Statement A: f has a limit at x = 2
To determine if the limit exists at x = 2, we need to evaluate the right and left limits and check if they are equal.
Right limit: lim(x->2+) [ (x^2 - 4) / (x - 2) ] = lim(x->2+) [(x + 2)] = 4
Left limit: lim(x->2-) [ (x^2 - 4) / (x - 2) ] = lim(x->2-) [(x + 2)] = 4

Since the right limit and left limit are equal (both equal to 4), the limit of f at x = 2 exists.

Statement B: f is continuous at x = 2
For f to be continuous at x = 2, the following conditions must be met:
1) f(x) is defined at x = 2 (It is defined as f(2) = 1)
2) The limit of f(x) as x approaches 2 must exist (Based on our evaluation above, the limit exists as 4)
3) The value of f(x) at x = 2 must be equal to the limit (f(2) = 1)

Since all three conditions are met, f is continuous at x = 2.

Statement C: f is differentiable at x = 2
To test for differentiability, we need to calculate the derivative at x = 2 by evaluating the limit of the difference quotient:
f'(2) = lim(h->0) [ f(2 + h) - f(2) ] / h

Let's calculate f'(2) using the definition of f(x):

f'(2) = lim(h->0) [ [((2 + h)^2 - 4) / (2 + h -2)] - 1 ] / h
= lim(h->0) [ (4 + 4h + h^2 - 4) / (h) ]
= lim(h->0) [ (4h + h^2) / (h) ]
= lim(h->0) [ 4 + h ]

Taking the limit as h approaches 0, we have f'(2) = 4.

Since f'(2) exists and is equal to 4, f is differentiable at x = 2.

Therefore, all three statements A, B, and C are true for the function f.

To determine which statement(s) about the function f is true, we need to analyze its behavior at x=2.

First, let's look at option A: "f has a limit at x=2." To check if this statement is true, we need to find the limit of f as x approaches 2 from both the left and right sides.

Taking the limit as x approaches 2 from the left side (x < 2), we substitute x = 2 - h into the function:

lim (h→0-) [(2 - h)^2 - 4] / (2 - h - 2)
= lim (h→0-) [(4 - 4h + h^2 - 4) / -h]
= lim (h→0-) [-4h / -h]
= lim (h→0-) 4
= 4

Taking the limit as x approaches 2 from the right side (x > 2), we substitute x = 2 + h into the function:

lim (h→0+) [(2 + h)^2 - 4] / (2 + h - 2)
= lim (h→0+) [(4 + 4h + h^2 - 4) / h]
= lim (h→0+) [4h / h]
= lim (h→0+) 4
= 4

Since the limit of f as x approaches 2 from both the left and right sides is 4, we can conclude that f has a limit at x=2. Therefore, option A is true.

Next, let's analyze option B: "f is continuous at x=2." To check if this statement is true, we need to determine if f is defined at x=2 and if the limit of f as x approaches 2 exists and is equal to f(2).

At x=2, the function is defined as f(2) = 1, which matches the provided condition. Additionally, we determined earlier that the limit of f as x approaches 2 is indeed 4. Since the limit equals f(2), we can conclude that f is continuous at x=2. Therefore, option B i true.

Finally, let's consider option C: "f is differentiable at x=2." To check if this statement is true, we need to examine the differentiability of f at x=2. For a function to be differentiable at a point, it must be continuous at that point.

Since we've already established that f is continuous at x=2, we can conclude that f is differentiable at x=2. Therefore, option C is true.

In summary, all three statements A, B, and C are true for the function f.