2.1.47 Calculation of limits
condition: Calculate the limits: a) \( \lim _{n \rightarrow \infty}\left(\frac{\sqrt{1+2+3+\cdots+n}}{2 n+1}-2 n\right) \), b) \( \lim _{n \rightarrow \infty}\left(\frac{\sqrt{n^{2}+1}+2}{\sqrt{n^{2}+1}+4}\right)^{2 n} \).
Calculation of various limits, using L'Hopital's rule, remarkable limits, power series, equivalent values of various other methods.