The definition of the derivative, question 27

A student uses the definition of the derivative to find $f'(2)$ for the function below, as shown. Find the first line that is wrong.

f(x) = x^{2} + 3

$f'(2) = \lim_{h \to 0} \frac{f(2 + h) - f(h)}{h}$
The definition of the derivative at x = 2.
$= \lim_{h \to 0} \frac{(2 + h)^{2} + 3 - \bigl(h^{2} + 3\bigr)}{h}$
Substitute into f.
$= \lim_{h \to 0} \frac{4 + 4h + h^{2} - h^{2}}{h}$
Expand the square; the constants cancel.
$= \lim_{h \to 0} \frac{4 + 4h}{h}$
The h² terms cancel.
$\text{The limit does not exist,} \quad \text{so } f'(2) \text{ does not exist.}$
The 4/h term blows up as h → 0.

How sure are you?

SureNot sureGuessing

My working

Saved in this browser only.

Corrected working