Question

What function f(x) and number a represent the derivative of the limit?

1) f(x) = x², a = 2
2) f(x) = sin(x), a = 0
3) f(x) = eˣ, a = 1
4) f(x) = ln(x), a = 1

Answer 1

The student's question pertains to finding the derivatives of specific functions at given points. The derivative represents the rate of change of a function, and each function's derivative can be evaluated at the specified points to obtain the rate of change at that location.

Step-by-step explanation:

The student's question seems to be asking about the concept of derivatives and limits in mathematics, specifically related to particular functions and a point at which to evaluate them. In this context, the function f(x) and the number a would be used to find the derivative of the function at the point x=a. Even though the student question does not explicitly define the term 'derivative of the limit,' it appears to be asking for us to consider the derivative of given functions at specific points.

To clarify the concepts:

• The derivative of a function represents the rate of change of the function at a particular point.
• A limit describes the behavior of a function as the input approaches a certain value.

Without knowing the full context or the complete question, it is challenging to provide an exact answer to the student's inquiry. However, I can explain how to find the derivative of the provided functions at the given points:

1. For f(x) = x², the derivative f'(x) is 2x. At a = 2, f'(2) = 4.
2. For f(x) = sin(x), the derivative f'(x) is cos(x). At a = 0, f'(0) = 1.
3. For f(x) = eˣ, the derivative f'(x) is eˣ. At a = 1, f'(1) = e.
4. For f(x) = ln(x), the derivative f'(x) is 1/x. At a = 1, f'(1) = 1.