(1) limn→∞(1+12n)n\lim_{n\to\infty}\left(1+\frac{1}{2n}\right)^nlimn→∞(1+2n1)n 2n = t とおくと =limn→±∞(1+1t)12t=e12=e=\lim_{n\to\pm\infty}\left(1+\frac{1}{t}\right)^{\frac{1}{2}t} = e^{\frac{1}{2}} = \sqrt{e}=limn→±∞(1+t1)21t=e21=e (2) limn→∞(1−2n)n\lim_{n\to\infty}\left(1-\frac{2}{n}\right)^nlimn→∞(1−n2)n −n2=t-\dfrac{n}{2} = t−2n=t とおくと =limn→±∞(1+1t)−2t=e−2=1e2=\lim_{n\to\pm\infty}\left(1+\frac{1}{t}\right)^{-2t} = e^{-2} = \frac{1}{e^2}=limn→±∞(1+t1)−2t=e−2=e21 (3) limn→∞(1−1n+1)n\lim_{n\to\infty}\left(1-\frac{1}{n+1}\right)^nlimn→∞(1−n+11)n −(n+1)=t-(n+1) = t−(n+1)=t とおくと =limn→±∞(1+1t)−t−1=limn→±∞(1+1t)−t×limn→±∞(1+1t)−1=e−1×(1+0)−1=1e=\lim_{n\to\pm\infty}\left(1+\frac{1}{t}\right)^{-t-1} = \lim_{n\to\pm\infty}\left(1+\frac{1}{t}\right)^{-t} \times \lim_{n\to\pm\infty}\left(1+\frac{1}{t}\right)^{-1} = e^{-1} \times (1+0)^{-1} = \frac{1}{e}=limn→±∞(1+t1)−t−1=limn→±∞(1+t1)−t×limn→±∞(1+t1)−1=e−1×(1+0)−1=e1