I want to improve my counting limits. I've found some difficult examples:
$\displaystyle\lim_{x \to +\infty}\left((x+1)^{1+\frac1x}-x^{1+\frac{1}{x+a}}\right)$
$\displaystyle\lim_{x\to +\infty}x^2(\arctan x - \frac{\pi}{2})+x$
$\displaystyle\lim_{x\to+\infty}\left( \sqrt[3]{x^3+x^2+x+1}-\sqrt{x^2+x+1}\cdot\frac{\ln(e^x+x)}{x} \right)$
$\displaystyle\lim_{x\to 0} \left(\frac{a^x-x\ln a}{b^x-x\ln b} \right)^{\frac{1}{x^2}}$
and I don't know how to touch them. I know: L'Hôpital's rule, Mean value theorem, Taylor's theorem but still don't have this skill. Can anybody help me?