Asosiy tarkibga o'tish

2.6-bo'lim : Cheksiz limitlar

4. \(\displaystyle g\left( x \right) = \frac{{x + 7}}{{{x^2} - 4}}\) baholash uchun,

  1. \(\mathop {\lim }\limits_{x \to {2^{\, - }}} g\left( x \right)\)
  2. \(\mathop {\lim }\limits_{x \to {2^{\, + }}} g\left( x \right)\)
  3. \(\mathop {\lim }\limits_{x \to 2} g\left( x \right)\)
a \(\mathop {\lim }\limits_{x \to {2^{\, - }}} g\left( x \right)\) Yechim

Keling, \(x \to {2^ - }\) uchun biz \(x < 2\)ni bilishimizni tan olishdan boshlaylik.

Surat uchun biz chegarada 9 ni olishimizni ko'rishimiz mumkin.

Endi maxrajga g'amxo'rlik qilaylik. Birinchidan, biz bilamizki, agar biz 2 dan kichik raqamni kvadratga aylantirsak (va -2 dan katta bo'lsa, bu erda bizda bor deb taxmin qilish mumkin, chunki biz chegarani bajaryapmiz) biz 4 dan kichik raqamni olamiz va shuning uchun chegarada bizda shunday bo'ladi:

\[{x^2} - 4 \to {0^ - }\]

Demak, chegarada surat ijobiy songa yaqinlashmoqda va maxraj borgan sari kichikroq manfiy sondir. Bo'lim katta manfiy son bo'lishi kerak yoki,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to {2^{\, - }}} \frac{{x + 7}}{{{x^2} - 4}} = - \infty }}\]

E'tibor bering, bu \(x = 2\) da vertikal asimptota mavjudligini ham anglatadi.


b \(\mathop {\lim }\limits_{x \to {2^{\, + }}} g\left( x \right)\) Yechim

Keling, \(x \to {2^ + }\) uchun biz \(x > 2\)ni bilishimizni tan olishdan boshlaylik.

Surat uchun biz chegarada 9 ni olishimizni ko'rishimiz mumkin.

Endi maxrajga g'amxo'rlik qilaylik. Birinchidan, biz bilamizki, agar biz 2 dan katta raqamni kvadratga aylantirsak, biz 4 dan katta sonni olamiz va shuning uchun chegarada biz shunday bo'ladi:

\[{x^2} - 4 \to {0^ + }\]

Demak, chegarada surat musbat songa yaqinlashmoqda va maxraj borgan sari kichikroq ijobiy sondir. Bunda ko'rsatkich ortib borayotgan katta musbat son bo'lishi kerak yoki,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to {2^{\, + }}} \frac{{x + 7}}{{{x^2} - 4}} = \infty }}\]

E'tibor bering, bu \(x = 2\) da vertikal asimptota mavjudligini ham anglatadi, biz uni birinchi qismdan bildik.


c \(\mathop {\lim }\limits_{x \to 2} g\left( x \right)\) Yechim

Bu holda biz dastlabki ikki qismdan shuni ko'rishimiz mumkinki,

\[\mathop {\lim }\limits_{x \to {2^{\, - }}} g\left( x \right) \ne \mathop {\lim }\limits_{x \to {2^{\, + }}} g\left( x \right)\]

va shuning uchun bizning asosiy chegara xususiyatlaridan biz \[\mathop {\lim }\limits_{x \to 2} g\left( x \right)\] ekanligini ko'rishimiz mumkin mavjud emas.

To'liqlik va ushbu muammoning javoblarini tekshirish uchun bu erda funktsiyaning qisqacha eskizi keltirilgan.

The graph of \(g\left( x \right) = \frac{{x + 7}}{{{x^2} - 4}}\). There are vertical asymptotes at \(x = - 2\) and \(x = 2\). On the far left the graph runs just below the x-axis and then rises steeply to positive infinity as \(x\) approaches -2 from the left. Between the two asymptotes the graph comes up from negative infinity, runs across just below the x-axis and then drops back down to negative infinity as \(x\) approaches 2 from the left. To the right of \(x = 2\) the graph comes down from positive infinity and flattens out just above the x-axis.

Ishonchimiz komilki, siz \(x = - 2\) da ushbu funktsiya uchun yana bir vertikal asimptota bo'lishini allaqachon payqagansiz. Amaliyot uchun siz ushbu nuqta uchun chegaralarni ham bajarishingiz mumkinligiga ishonch hosil qilishni xohlashingiz mumkin.