f(x) = 3x^2 Obtain $$ ext{lim}_{h o 0} \frac{f(x + h) - f(x)}{h}$$ Circle your answer. - AQA - A-Level Maths Mechanics - Question 3 - 2021 - Paper 3

To calculate the limit, we start by substituting the expression for $f(x)$ into the limit definition:

1. Calculate $f(x + h)$: $f(x + h) = 3(x + h)^2 = 3(x^2 + 2xh + h^2) = 3x^2 + 6xh + 3h^2$

2. Substitute into the limit:

   $\frac{f(x + h) - f(x)}{h} = \frac{(3x^2 + 6xh + 3h^2) - 3x^2}{h} = \frac{6xh + 3h^2}{h}$

   Simplifying gives: $6x + 3h$

3. Now, take the limit as $h$ approaches 0:

   $\text{lim}_{h o 0} (6x + 3h) = 6x$

Thus, the final answer is:

$6x$