Does f'(x) approach 0 if f(x) approaches 0 as x→∞?

[calculus_jy]

This questions has been bothering me for quiet a while

Does $$lim_{x\rightarrow \infty} f(x)=0 \Rightarrow lim_{x\rightarrow \infty} f'(x)=0$$ if so can you please post a proof because its giving me a headache. If not would it please be possible for you to give me a counter example.

Many thanks

 

[least_action]

This questions has been bothering me for quiet a while

Does $$lim_{x\rightarrow \infty} f(x)=0 \Rightarrow lim_{x\rightarrow \infty} f'(x)=0$$ if so can you please post a proof because its giving me a headache. If not would it please be possible for you to give me a counter example.

Many thanks

$$\frac{sin(x^2)}{x}$$