Btukfyl

3 $begingroup$ I got stuck on two exercises below $$ limlimits_{xrightarrow +infty} left(frac{2}{pi} arctan x right)^x \ lim_{xrightarrow 3^+} frac{cos x ln(x-3)}{ln(e^x-e^3)} $$ For the first one , let $y=(frac{2}{pi} arctan x )^x $ , so $ln y =xln (frac{2}{pi} arctan x )$ , the right part is $infty cdot 0$ type, but seemly, the L 'hopital's rule is useless. PS: I know the $infty cdot 0$ can be become to $frac{infty}{infty}$ or $frac{0}{0}$ . But when I use the L 'hopital's rule to the $frac{infty}{infty}$ or $frac{0}{0}$ the calculation is complex and useless. For the second one , it is $frac{infty}{infty}$ type, also useless the L 'hopital's rule is. How to calculate it ? limits ...