Question

Find, from the first principles, the gradient function of:

Answer 1

F(x+h) = x+h, and f(x) = x

So, your limit is:

$\lim_{h \to 0} \frac{x+h - x}{h} = \lim_{h \to 0} \frac{h}{h}$

Applying l'hopital's rule, 

$\lim_{h \to 0} \frac{h}{h} = \lim_{h \to 0} \frac{1}{1} = 1$

Giving a gradient of 1.

Answer 2

$\begin{aligned}f'(x) &= \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h} \\&= \lim_{h \to 0} \dfrac{(x+h) - x}{h} \\&=\lim_{h \to 0} \dfrac{h}{h} \\&=\lim_{h \to 0} (1) && (\text{\footnotesize since h/h = 1 for h\ne 0 })\\&= 1\end{aligned}$

h/h = 1 is valid for h ≠ 0. The limit does not care about h at 0; it only cares about the values around it.