How to prove that $\lim_{x\to \infty} x/2^x = 0$

Question

I need to prove that $\lim_{x\to \infty} x/2^x = 0$

I'm not sure I did it right:

I applied L'ôpital's rule and obtainded: $\lim_{x\to \infty} \dfrac{1}{2^x\ln2}$

and this is equal to $\dfrac{1}{\ln2} . \lim_{x\to \infty} \dfrac{1}{2^x}$

so $\lim_{x\to \infty} x/2^x = 0$

Is this correct?

Answer 1

Yes, your solution is correct, and your approach is just fine.

L'Hopital is perfectly appropriate to apply here, given the indeterminate evaluation of $$\lim_{x\to \infty} \dfrac x{2^x}$$