2.2-bo'lim : Limit tushunchasi (The Limit)
3. Funksiya: \(\displaystyle g\left( \theta\right) = \frac{{\sin \left( {7\theta } \right)}}{\theta }\) uchun quyidagi savollarning har biriga javob bering.
- Funksiyani quyidagi qiymatlarda hisoblang: \(\theta \) (kamida 8 ta o'nlik xonagacha aniqlikda hisoblang). Hisoblashlar uchun kalkulyatoringiz radian rejimiga o'rnatilganligiga ishonch hosil qiling.
- 0.5
- 0.1
- 0.01
- 0.001
- 0.0001
- -0.5
- -0.1
- -0.01
- -0.001
- -0.0001
- (a) bandidagi ma'lumotlardan foydalanib, limit qiymatini baholang (taxmin qiling): \(\displaystyle \mathop {\lim }\limits_{\theta\to \,0} \frac{{\sin \left( {7\theta } \right)}}{\theta }\).
- 0.5
- 0.1
- 0.01
- 0.001
- 0.0001
- -0.5
- -0.1
- -0.01
- -0.001
- -0.0001
Yechim
Quyida funksiyaning berilgan nuqtalardagi 8 ta o'nlik xonagacha aniqlikdagi qiymatlar jadvali keltirilgan.
| \(\theta\) | \(g(\theta)\) | \(\theta\) | \(g(\theta)\) |
|---|---|---|---|
| 0.5 | -0.70156646 | -0.5 | -0.70156646 |
| 0.1 | 6.44217687 | -0.1 | 6.44217687 |
| 0.01 | 6.99428473 | -0.01 | 6.99428473 |
| 0.001 | 6.99994283 | -0.001 | 6.99994283 |
| 0.0001 | 6.99999943 | -0.0001 | 6.99999943 |
b (a) bandidagi ma'lumotlardan foydalanib, limit qiymatini baholang (taxmin qiling): \(\displaystyle \mathop {\lim }\limits_{\theta\to \,0} \frac{{\sin \left( {7\theta } \right)}}{\theta }\). Yechim
Yuqoridagi qiymatlar jadvalidan ko'rinib turibdiki, biz quyidagicha baholashimiz (taxmin qilishimiz) mumkin:
\[\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \left( {7\theta } \right)}}{\theta } = 7\]