Let f be a function such that the limit as h approaches 0 [(f(5+h)-f(5))/h]=4. Which of the following must be true?
I. f(5)=4
II. f'(5)=4
III. f is continuous at x=5
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
the value given is the definition of f'(5).
So, II and III are true, making E the correct choice.
Well, it seems like we've got a math question here, so I'll do my best to clown around with this one!
Let's start by analyzing the options, one by one:
I. f(5) = 4
Hmm, this says that f(5) must equal 4. Well, the limit given only tells us about the derivative at x=5, not the actual value of f(5). So, sorry to rain on this option's parade, but it doesn't have to be true.
II. f'(5) = 4
Ah, now this option claims that the derivative of f at x=5 is 4. Since the limit involving f'(5) is given as 4, it's possible for this to be true. So, we've got a contender here!
III. f is continuous at x=5
This option suggests that f must be continuous at x=5. However, the given limit gives us information about the derivative, not continuity. So, this option doesn't necessarily have to be true.
So, after all the clowning around, we can see that the correct answer is:
B. II only
Because f'(5) = 4 can be concluded from the given limit. Keep the clown-party going!
To determine which of the statements are true, let's analyze each one separately:
I. f(5)=4
To determine the value of f(5), we can rewrite the limit expression in terms of f(x) as follows:
lim(h->0) [(f(5+h)-f(5))/h] = 4
When h approaches 0, f(5+h) approaches f(5). Therefore, we can substitute f(5) with a constant 'c':
lim(h->0) [(c-f(5))/h] = 4
By simplifying further, we can multiply through by 'h' to get:
c - f(5) = 4h
As h approaches 0, the right-hand side approaches 0. Thus, c - f(5) must also approach 0. This implies that c = f(5).
Therefore, f(5) must equal 4. Hence, statement I is true.
II. f'(5) = 4
The expression [(f(5+h)-f(5))/h] represents the average rate of change of f(x) between x = 5 and x = 5+h. The fact that this limit equals 4 suggests that the function is increasing at x = 5.
However, this does not necessarily mean that the derivative of f at x = 5 is equal to 4. The derivative represents the instantaneous rate of change, and it may differ from the average rate of change. Therefore, statement II is not necessarily true.
III. f is continuous at x = 5
We cannot determine the continuity of f at x = 5 based solely on the given information. The fact that the limit exists does not guarantee that the function is continuous at x = 5. Therefore, statement III is not necessarily true.
In conclusion, statement I is the only one that must be true. Hence, the answer is A. I only.
To determine which of the statements must be true, we need to analyze the given limit and its implications.
Let's examine the limit as h approaches 0: [(f(5+h) - f(5)) / h] = 4
This limit represents the derivative of f(x) at x = 5. In other words, it tells us the slope of the tangent line to the graph of f(x) at x = 5.
Statement I states that f(5) = 4. To verify if this is true, we can't solely rely on the given limit. Instead, we need to consider additional information, such as the value of f(5) or the behavior of f(x) near x = 5. Thus, Statement I cannot be determined solely by the given limit.
Statement II states that f'(5) = 4. The given limit represents the derivative of f(x) at x = 5, so Statement II is directly implied by the given limit. Therefore, Statement II must be true.
Statement III states that f is continuous at x = 5. Although the given limit provides information about the derivative of f(x) at x = 5, it does not indicate anything about the continuity of f(x) at that point. Thus, Statement III cannot be determined solely by the given limit.
To summarize, Statement II ("f'(5) = 4") must be true based on the given limit, while Statements I ("f(5) = 4") and III ("f is continuous at x = 5") cannot be determined solely by the given limit.
Therefore, the correct answer is B. II only.