How do you find f'(2) using the limit definition given f(x)=5x-3

Answer 1

You want to evaluate the limit as h--> 0 of (f(a+h) - f(a))/h where a = 2 and f(x) = 5x-3 and f(2) = 7

Lim as h-->0 of (5(h+2)-3 - 7)/h = Lim as h--> 0 5h/h = 5

Answer 2

f'(x) = lim_h→0 [(f(x+h) - f(x)) / h] = lim_h→0 [(5(x+h) - 3 - (5x - 3)) / h] = lim_h→0 [5h / h] = lim_h→0(5) = 5

So, for any x, f'(x) = 5. Thus, f'(2) = 5.