Question

Solve using the limit definition, limit as h approaches 0:

Answer 1

limit formula is: $\frac{f(x + h) - f(x)}{h}$

f(x + h)= (x + h)³ - 3(x + h) + 7

          = x³ + 3x²h + 3xh² + h³ - 3x - 3h + 7

f(x) = x³ - 3x + 7

f(x + h) - f(x)= (a³ + 3a²h + 3ah² + h³ - 3a - 3h + 7) - (a³ - 3a + 7)

                   = 3x²h + 3xh² + h³  - 3h

$\frac{f(x + h) - f(x)}{h}$ = $\frac{3x^2h + 3xh^2 + h^3 - 3h}{h} = \frac{h(3x^2 + 3xh + h^2 - 3)}{h}$ =  3x² + 3xh + h² - 3

as h approaches 0:  3x² + 3x(0) + (0)² - 3   = 3x² - 3