Calculate $\lim_{x\to\infty} (\sin\frac{1}{x}+\cos\frac{1}{x})^x$ without using l'Hopital's rule.
I attempted pulling something out to get a limit that resembles $e$, this gave me:
$$\sin^x\frac{1}{x}(1+\frac{1}{\tan\frac{1}{x}})^x=\sin^x\frac{1}{x}\bigg[(1+\frac{1}{\tan\frac{1}{x}})^{\tan\frac{1}{x}}\bigg]^{\frac{x}{\tan\frac{1}{x}}}$$
But then the first part goes to $0$ and the exponent goes to infinity, and I'm kinda stumped.