Final answer:
The derivative of the function f(x) = 7x - 3, found by the limit process, is f'(x) = 7. This is determined by applying the definition of the derivative, simplifying the algebraic expression, and recognizing that the derivative of a constant is zero.
Step-by-step explanation:
To find the derivative of the function f(x) = 7x - 3 by the limit process, we employ the definition of the derivative, which is:
f'(x) = lim (h → 0) [(f(x+h) - f(x))/h]
Applying this definition to our function, we get:
f'(x) = lim (h → 0) [((7(x+h) - 3) - (7x - 3))/h]
Simplifying the expression inside the limit, we have:
f'(x) = lim (h → 0) [(7x + 7h - 3 - 7x + 3)/h]
The terms 7x and -7x cancel out, and the constants -3 and +3 cancel each other as well:
f'(x) = lim (h → 0) [7h/h]
Now, we can simplify by canceling h:
f'(x) = lim (h → 0) [7]
Since 7 is a constant and does not depend on h, we can conclude:
f'(x) = 7