1. Muammo: Quyidagi limitni hisoblang: $$\lim_{n \to \infty} \frac{n^n}{\left[(n+1)!\right]^2}$$ 2. Formulalar va qoidalar: Faktorial ta'rifi \( (n+1)! = 1 \cdot 2 \cdot 3 \cdots (n+1) \) va limitni hisoblashda Stirling formulasi yoki taqqoslash usullaridan foydalanish mumkin. 3. Hisoblash: \( (n+1)! = (n+1) \cdot n! \), shuning uchun $$\left[(n+1)!\right]^2 = ((n+1)!)^2 = (n+1)^2 (n!)^2$$ 4. Shunday qilib, ifoda: $$\frac{n^n}{((n+1)!)^2} = \frac{n^n}{(n+1)^2 (n!)^2}$$ 5. Endi \(n!\) ni Stirling formulasiga yaqinlashtiramiz: $$n! \approx \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n$$ 6. Shunday qilib, $$ (n!)^2 \approx 2 \pi n \left(\frac{n}{e}\right)^{2n} $$ 7. Ifodani almashtiramiz: $$\frac{n^n}{(n+1)^2 \cdot 2 \pi n \left(\frac{n}{e}\right)^{2n}} = \frac{n^n}{(n+1)^2 \cdot 2 \pi n} \cdot \frac{e^{2n}}{n^{2n}} = \frac{e^{2n}}{(n+1)^2 \cdot 2 \pi n} \cdot \frac{n^n}{n^{2n}}$$ 8. Soddalashtiramiz: $$\frac{n^n}{n^{2n}} = n^{-n}$$ Shunday qilib, $$\frac{e^{2n}}{(n+1)^2 \cdot 2 \pi n} \cdot n^{-n} = \frac{e^{2n}}{(n+1)^2 \cdot 2 \pi n} \cdot \frac{1}{n^n}$$ 9. Limitga qarasak, \(n^n\) juda tez o'sadi, shuning uchun \(\frac{1}{n^n} \to 0\) juda tez kamayadi va butun ifoda 0 ga yaqinlashadi. 10. Natija: $$\lim_{n \to \infty} \frac{n^n}{\left[(n+1)!\right]^2} = 0$$ Bu limit 0 ga teng.