1.次の数列の極限を求めなさい。
(1)\(\displaystyle \lim_{n\to\infty}\frac{3^{n+1}}{2^n+3^n}\)
\(\displaystyle =\lim_{n\to\infty}\frac{3}{(\frac{2}{3})^n+1}\)
\(\displaystyle =3\)
(2)\(\displaystyle \lim_{n\to\infty}\frac{3^n+2}{2^n+1}\)
\(\displaystyle =\lim_{n\to\infty}\frac{3^n(1+\frac{2}{3^n})}{2^n(1+\frac{1}{2^n})}\)
\(\displaystyle =\lim_{n\to\infty}\left(\frac{3}{2}\right)^n\frac{1+\frac{2}{3^n}}{1+\frac{1}{2^n}}\)
\(\displaystyle =\infty\)
(3)\(\displaystyle \lim_{n\to\infty}\frac{2^{2n}+3^n}{4^{n+1}+2^{n+1}}\)
\(\displaystyle =\lim_{n\to\infty}\frac{4^n+3^n}{4^{n+1}+2^{n+1}}\)
\(\displaystyle =\lim_{n\to\infty}\frac{1+(\frac{3}{4})^n}{4+2(\frac{1}{2})^n}\)
\(\displaystyle =\frac{1}{4}\)