Let f be a real value function and x Î Df
then
0
lim ( ) ( )
h
f x h f x
® h
+ -
when it exists is
called
A) The derivative of f at a
B) The derivative of f at h
C) The derivative of f at x
D) The derivative of f at x = h
Can't read your funky fonts, but since
there are no a's involved, and
there x's and h's,
I'd say (C)
The limit that you have given represents the definition of the derivative of a function. To find the answer to the question, we need to understand the notation and apply it correctly.
The notation "lim" represents the limit as a certain variable approaches a particular value. In this case, it represents the limit as "h" approaches 0. So, we are looking for the limit of the expression (f(x+h) - f(x))/h as h approaches 0.
Now, let's go through each option to determine the correct answer:
A) The derivative of f at a: This option is not applicable to the given situation because there is no "a" involved in the expression or limit.
B) The derivative of f at h: This option is not applicable either because "h" is not an independent variable of the function f(x). It is a small change in the input value of x.
C) The derivative of f at x: This is the correct answer. The expression (f(x+h) - f(x))/h represents the rate of change of the function f at x, which is the definition of the derivative of f at x.
D) The derivative of f at x = h: This option is incorrect because x and h represent different variables. "h" is not equal to "x".
Therefore, the correct answer is C) The derivative of f at x.