\[ \begin{split} &\lim_{n \to \infty} 2^n\operatorname{tg} 2^{-n}=\\[6pt] &=lim_{n \to \infty}\dfrac{2^n\sin2^{-n}}{\cos 2^{-n}}=\\[6pt] &=lim_{n \to \infty}\dfrac{\sin2^{-n}}{2^{-n}\cos 2^{-n}}=\\[6pt] &=\begin{vmatrix} 2^{-n}=t; && \dfrac{1}{2^n}=t;\\ n \to \infty &to& t \to 0 \end{vmatrix}=\\[6pt] &=lim_{t \to 0}\frac{\sin t}{t\cdot \cos t} = 1 \end{split} \]