Problem 816 (difficulty: 3/10)

\(\displaystyle \lim_{x\to0}\frac{\sin 3x}{\tg 5x}=? \quad \lim_{x\to0}\frac{\log\cos ax}{\log\ch bx}=? \quad \lim_{x\to0}\left(\frac{\sin x}x\right)^{x^{-2}}=? \quad \lim_{x\to1}\left((x-1)\tg\frac{\pi x}2\right)=? \quad \lim_{x\to\infty}\frac{\sin\ln x}{x} = ? \)

Can we use the L'Hospital rule? Can we use the definition of the derivative at \(\displaystyle 0\) (or \(\displaystyle 1\))?

Give me another random problem!