Please help me prove LIM (1 + 1 / N) ^ n = e, thank you!

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• 1. lim n→∞ [(n!)^1/n]/n = 1/e Our advanced mathematics teacher uses the idea of LIM an = a, Lim BN = (a1 + A2 +... + an) / N = a
• 2. When n - > + ∞, Lim [e ^ n / N ^ 2] =? And prove
• 3. Lim X - > 0, the limit of sin (1 / x)? Lim X - > 0, the limit of X * (sin (1 / x))?
• 4. The limit of LIM [(x times of a + x times of B + x times of C) / 3] is X - > 0 X is close to 0, [(x times of a + x times of B + x times of C) / 3] the limit of x-th power is wrong, it is x-th power, and there are fewer times
• 5. LIM (x tends to 3) 1 / x = 1 / 3
• 6. Find the limit of LIM (x → 1) [(^ 3 √ x-1) / (x-1)]
• 7. Find the limit of LIM (x → 0) [(a ^ x-1) / x] a>0,a≠1
• 8. Find the limit of LIM (x →∞) [1 - (3 / x)] ^ X
• 9. Find the limit of LIM (x → 0) [(2 ^ x + 3 ^ x) / 2] ^ (1 / 2) Sorry, it's LIM (x → 0) [(2 ^ x + 3 ^ x) / 2] ^ (1 / x)
• 10. Find limit Lim X - > infinite ((2x-1) ^ 30 (X-2) ^ 5) / (2x + 1) ^ 35
• 11. Lim（(1²＋2²＋…… ＋n²)/n³）n→∞＝
• 12. Prove LIM (n →∞) (1 + 1 / N & # 178;) = 1 by definition
• 13. Lim √ (n & # 178; - A & # 178;) / N = 1 proved by definition N approaches positive infinity
• 14. Proof of mathematical limit: LIM (n-positive infinity) [(- 1) ^ n / N ^ 2] = 0
• 15. X →∞, Lim [(x ^ 2 / 1 + x) + ax-b] = 1, find a, B
• 16. Lim X - > 0 + X / (1-cosx) ^ 1 / 2 why is the answer 2
• 17. Find the limit LIM (x → 1) (x - N - 1) / (x-1) =? N is a positive integer
• 18. Find function limit LIM (n-1) * 2 ^ n, n tends to be positive infinity
• 19. Finding function limit LIM (Sinn) / n Finding function limit LIM (n!) / N ＾ n N tends to infinity Why?
• 20. lim n->0 (1+3n/1)^3n = ? lim n->0 (1+3n/1)^3n = ?