Question

Solve the limit as h approaches 0 of 1/h times the integral from x to x + h of f(x).

a) 0
b) f(x)
c) h
d) 1

Answer 1

The limit as h approaches 0 of 1/h times the integral from x to x + h of f(x) is f(x), based on the application of the fundamental theorem of calculus.

Step-by-step explanation:

The student asked to solve the limit as h approaches 0 of 1/h times the integral from x to x + h of f(x). This is a problem related to the fundamental theorem of calculus and involves concepts such as limits and integrals.

To solve this, we can recognize that as h approaches 0, the function f(x) over the interval from x to x + h becomes constant because the width of the interval is shrinking to 0. Applying the fundamental theorem of calculus, we have:

1. The integral of a constant f(x) over an interval [x, x+h] is simply f(x) times h.
2. Therefore, 1/h times the integral from x to x + h of f(x) is f(x).
3. Finally, we apply the limit as h approaches 0, which leaves us with f(x) since f(x) does not depend on h.

Thus, the answer is b) f(x).