Let f be a function such that lim as h approaches 0 f(2+h)-f(2)/h=0 Which of the following must be true

1. f is continuous at x=2
2. f is differnetiable at x=2
3. f has a horizontal tangent line at x=2
4. f travels through the origin

1 2 and 3 are correct but how should I explain why

the limit exists, and it is in fact the derivative

2: This is a limit definition of a derivative, specifically of f(x) for x=2. The limit is definable as 0, so f'(2)=0; that is, f is differentiable at x=2.

1: A function can only be differentiable at a point if it is continuous at that point, so f is continuous at x=2.
3: f'(2)=0, which is a horizontal tangent (the slope of f is 0, or horizontal).

4: Incorrect as there is no indication that f passes through (0, 0).

Hope that helps!

Well, let's bring in the clown to explain this one!

🀑 Hello there, dear student! Let's have some fun explaining why options 1, 2, and 3 are true based on the given information.

First, if the limit as h approaches 0 of [f(2+h) - f(2)]/h is equal to 0, it means that the function f is continuous at x = 2. Why? Because to have a limit exist, we need both the left-hand limit and the right-hand limit to be the same. This suggests that there are no jumps or discontinuities around x = 2, indicating continuity.

Now, if f is continuous at x = 2, it also means that f is differentiable at x = 2. Why? Because differentiability is a stronger condition than continuity. If a function is continuous at a point, it must also be differentiable at that point. Think of it this way: if you can smoothly connect the dots close to x = 2, then you can also find the slope at that point.

As for option 3, the fact that the limit of [f(2+h) - f(2)]/h as h approaches 0 is 0 indicates that the tangent line to the graph of f at x = 2 is horizontal. Why? Because for the slope to be 0, the tangent line must be parallel to the x-axis, which makes it horizontal.

So, options 1, 2, and 3 are all true because of the given limit! Isn't math funny sometimes? 🀑

To determine which of the given statements must be true, let's go through them one by one and explain why they are correct:

1. f is continuous at x=2:
To determine if a function is continuous at a specific point, it must satisfy three conditions:
(a) The function must be defined at that point (f(2) exists).
(b) The limit of the function as x approaches that point must exist (lim as x approaches 2 f(x) exists).
(c) The limit of the function as x approaches that point must be equal to the value of the function at that point (lim as x approaches 2 f(x) = f(2)).

From the given information, we have the limit as h approaches 0 of f(2+h)-f(2)/h = 0. This implies that the limit of f(x) as x approaches 2 exists. Since f(2) is defined and the limit equals f(2), all three conditions for continuity are satisfied. Therefore, f is continuous at x=2.

2. f is differentiable at x=2:
To determine if a function is differentiable at a specific point, the function must be continuous at that point, and the derivative of the function must exist at that point.

From the previous explanation, we've established that f is continuous at x=2. Now, we need to check if the derivative of f exists at x=2. Given the expression lim as h approaches 0 f(2+h)-f(2)/h = 0, the left-hand side corresponds to the definition of the derivative of f at x=2. Since this limit equals zero, we can conclude that f is differentiable at x=2.

3. f has a horizontal tangent line at x=2:
A function has a horizontal tangent line at a specific point if its derivative at that point equals zero.

From the previous explanation, we've determined that f is differentiable at x=2. Since the derivative of f exists at this point and is equal to zero (as shown by the previous limit), we can conclude that f has a horizontal tangent line at x=2.

4. f travels through the origin:
From the given information, we cannot determine whether or not f passes through the origin (0,0). The limit expression provided does not give us any direct information about the y-values or intercepts of the function. Thus, we cannot conclude that f travels through the origin based on the given information.

In summary, based on the given information, statements 1, 2, and 3 are true, while statement 4 cannot be determined.