The following function is continuous but cannot be differentiated at x=1 $$ f(x) = \begin{cases} \frac13 x^3 + \frac12 x^2 &&\text{if } x\ge 1 \\ x- \frac16 &&\text{if } x < 1 \end{cases} $$
If we take a sequence $\lim h_n = 0$ and suppose $h_n$ > $0$ then $$\lim \frac{f(1+h_n) - f(1)} {h_n} $$
I don't know where to from here, if I plug in values from $f(x)$ I will get the same limit from each equation.
Thank you.
$h_n$ positive accounts for only one side of the function. You must also consider the case where $h_n$ is negative, and compute the derivative (limit) under this assumption. When $h_n<0$ the definition of $f$ changes, and you should find that you get a different limit from the $h_n>0$ case, which shows non-differentiability.