2.7-bo'lim : Infinity chegaralari, I qism
4. \(\displaystyle f\left( x \right) = \frac{{3{x^7} - 4{x^2} + 1}}{{5 - 10{x^2}}}\) uchun quyidagi savollarning har biriga javob bering.
- \(\mathop {\lim }\limits_{x \to \, - \infty } f\left( x \right)\) ni baholang.
- (b) \(\mathop {\lim }\limits_{x \to \,\infty } f\left( x \right)\) ni baholang.
- Funktsiya uchun har qanday gorizontal asimptotalarning tenglama(lar)ini yozing.
a \(\mathop {\lim }\limits_{x \to \, - \infty } f\left( x \right)\) ni baholang. Yechim
Buni amalga oshirish uchun biz qilishimiz kerak bo'lgan yagona narsa \(x\) ning har ikkala maxrajdan ham maxrajdagi eng katta kuchini hisobga olishdir. va surat. Keyin biz qilishimiz kerak bo'lgan yagona narsa asosiy chegara xususiyatlaridan foydalanishdir 1-fakt limitni hisoblash uchun ushbu bo'limdan.
\[\mathop {\lim }\limits_{x \to \, - \infty } \frac{{3{x^7} - 4{x^2} + 1}}{{5 - 10{x^2}}} = \mathop {\lim }\limits_{x \to \, - \infty } \frac{{{x^2}\left( {3{x^5} - 4 + \frac{1}{{{x^2}}}} \right)}}{{{x^2}\left( {\frac{5}{{{x^2}}} - 10} \right)}} = \mathop {\lim }\limits_{x \to \, - \infty } \frac{{3{x^5} - 4 + \frac{1}{{{x^2}}}}}{{\frac{5}{{{x^2}}} - 10}} = \frac{{ - \infty }}{{ - 10}} = \require{bbox} \bbox[2pt,border:1px solid black]{\infty }\]b \(\mathop {\lim }\limits_{x \to \,\infty } f\left( x \right)\) ni baholang. Yechim
Ushbu qism uchun biz birinchi qismda qilgan barcha matematik manipulyatsiyalar chegaraning o'ziga bog'liq emas edi va shuning uchun bu erda qayta ishlash kerak emas. Biroq, ularni qo'shish juda oson, shuning uchun biz davom etamiz va ularni qo'shamiz.
\[\mathop {\lim }\limits_{x \to \,\infty } \frac{{3{x^7} - 4{x^2} + 1}}{{5 - 10{x^2}}} = \mathop {\lim }\limits_{x \to \,\infty } \frac{{{x^2}\left( {3{x^5} - 4 + \frac{1}{{{x^2}}}} \right)}}{{{x^2}\left( {\frac{5}{{{x^2}}} - 10} \right)}} = \mathop {\lim }\limits_{x \to \,\infty } \frac{{3{x^5} - 4 + \frac{1}{{{x^2}}}}}{{\frac{5}{{{x^2}}} - 10}} = \frac{\infty }{{ - 10}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \infty }}\]c Funktsiya uchun har qanday gorizontal asimptotalarning tenglama(lar)ini yozing. Yechim
Biz bilamizki, \(x \to - \infty \) uchun gorizontal asimptota bo'ladi, agar \(\mathop {\lim }\limits_{x \to \, - \infty } f\left( x \right)\) mavjud bo'lsa va cheklangan son bo'lsa. Xuddi shunday, agar \(\mathop {\lim }\limits_{x \to \,\infty } f\left( x \right)\) mavjud bo'lsa va cheklangan son bo'lsa, bizda \(x \to \infty \) uchun gorizontal asimptota bo'ladi.
Shuning uchun, dastlabki ikki qismdan biz ushbu funktsiyaga ega bo'lishini ko'rishimiz mumkin gorizontal asimptotlar yo'q chunki ikkita chegaraning hech biri chekli emas.