advanced limits exercise with trigonometry

Let's generalize!

Consider a differentiable map $f$. We want to evaluate

$$\lim\limits_{h \to 0} \frac{f(a+2h)-2 f(a+h) +f(a)}{h}$$

You have

$$ \frac{f(a+2h)-2f(a+h)+f (a)}{h} = \frac{f(a+2h)-f(a)}{h} - \frac{f(a+h)-f( a)}{h}- \frac{f(a+h)-f(a)}{h} $$

For $h \to 0$, the first term of the RHS converges to $2f^\prime(a)$, the second and the third one to $f^\prime(a)$. Hence the limit is equal to zero.

Take $\tan$ for $f$. The limit is equal to zero.