Let f be a function defined for all real numbers. Which of the following statements must be true about f? Which might be true? Which must be false? Justify your answers.

(a) lim of f(x) as x approaches a = f(a)

(b) If the lim of f(x)/x as x approaches 0 = 2, then f(0)=0.

(c) If the lim of f(x)/x as x approaches 0 =1, then the lim of f(x) as x approaches 0 =0

(d) If the limit of f(x)-f(0) all over x as x approaches 0=3, then f'(0)=3.

Let's analyze each statement one by one:

(a) The statement (a) is stating that the limit of f(x) as x approaches a is equal to f(a). This statement is the definition of continuity. So, if a function is continuous at a particular point a, then this statement must be true. However, it is important to note that not all functions are continuous at every point, so this statement might not be true for all functions.

(b) The statement (b) is saying that if the limit of f(x)/x as x approaches 0 is equal to 2, then f(0) must be equal to 0. This is not a true statement in general. Consider the function f(x) = 2x. In this case, the limit of f(x)/x as x approaches 0 is 2, but f(0) = 0.

(c) The statement (c) is stating that if the limit of f(x)/x as x approaches 0 is equal to 1, then the limit of f(x) as x approaches 0 must be equal to 0. This statement is true. It follows from the fact that if the limit of f(x)/x as x approaches 0 exists and is equal to a finite number, then the limit of f(x) as x approaches 0 also exists and is equal to the same number multiplied by 0, which is 0.

(d) The statement (d) is suggesting that if the limit of (f(x) - f(0))/x as x approaches 0 is equal to 3, then f'(0) must be equal to 3. This is not necessarily true. The limit (f(x) - f(0))/x as x approaches 0 represents the derivative of f(x) at x = 0. However, the limit only gives the slope of the tangent line at x = 0, it doesn't provide information about the value of the function itself. So, f'(0) could be any value, not necessarily 3.

To summarize:
(a) Must be true if the function is continuous at a.
(b) Might be true, but not necessarily.
(c) Must be true.
(d) Must be false.

To determine which statements must be true, might be true, or must be false about the function f, let's go through each statement one by one.

(a) Statement: The limit of f(x) as x approaches a is equal to f(a).
This statement is known as the limit definition of continuity. If f is continuous at the point a, then this statement must be true. However, if f is not continuous at a, this statement can be false. Therefore, the statement might be true but does not necessarily have to be true.

(b) Statement: If the limit of f(x)/x as x approaches 0 is equal to 2, then f(0) = 0.
To evaluate if this statement is true, we can take the limit itself. If the limit of f(x)/x as x approaches 0 is equal to 2, it means that as x approaches 0, the function f(x) approaches 2x. In this case, f(0) must be equal to 0 for f to be continuous at x = 0 by the limit definition of continuity. Therefore, this statement must be true.

(c) Statement: If the limit of f(x)/x as x approaches 0 is equal to 1, then the limit of f(x) as x approaches 0 is equal to 0.
Again, we can consider the limit itself. If the limit of f(x)/x as x approaches 0 is equal to 1, it means that as x approaches 0, the function f(x) approaches x. In this case, the limit of f(x) as x approaches 0 must also be equal to 0 for f to be continuous at x = 0 by the limit definition of continuity. Therefore, this statement must be true.

(d) Statement: If the limit of (f(x) - f(0))/x as x approaches 0 is equal to 3, then f'(0) = 3.
This statement is about the derivative of f. If the limit of (f(x) - f(0))/x as x approaches 0 is equal to 3, it means that the derivative of f at x = 0 is equal to 3. However, this does not necessarily imply that f'(0) is equal to 3. The limit of the difference quotient is used to define the derivative, but it does not guarantee that the value of the derivative at that point will be the same. Therefore, this statement might be true but does not necessarily have to be true.

In summary:
(a) Might be true.
(b) Must be true.
(c) Must be true.
(d) Might be true.

(a) might be true (sin(0)/0 not defined)

(b) might be true (f(0) might not be defined)
(c) must be true
(d) must be true - that's the definition of f'