So the Squeeze Theorem says $\displaystyle \lim_{x\to\infty}\frac{\sin(x^2)}{x}=0$
The derivative of $\displaystyle \frac{\sin x}{x^2}$ is $\displaystyle \frac{\cos x}{x^2}-\frac{2\sin x}{x^3}$, which clearly goes to $\displaystyle 0$ as $\displaystyle x\to\infty$.
This is a mess and everyone understands what she/he wants : I meant that in my example $\displaystyle \frac{sin x}{x}$ I forgot to square the denominator when differentiating and thus got a wrong derivative.
This is a mess and everyone understands what she/he wants : I meant that in my example $\displaystyle \frac{sin x}{x}$ I forgot to square the denominator when differentiating and thus got a wrong derivative.