Mutlaq qiymat funksiyasining ta'rifini eslang.

\[\left| p \right| = \left\{ {\begin{array}{*{20}{c}}p&{p \ge 0}\\{ - p}&{p < 0}\end{array}} \right.\]

Shunday qilib, \(t = - 1\) da mutlaq qiymat ichidagi funktsiya nolga teng bo'lgani uchun biz buni ko'rishimiz mumkin,

\[\left| {t + 1} \right| = \left\{ {\begin{array}{*{20}{c}}{t + 1}&{t \ge - 1}\\{ - \left( {t + 1} \right)}&{t < - 1}\end{array}} \right.\]

Bu shuni anglatadiki, bizdan "kesish" nuqtasidagi chegarani bo'lak-bo'lak funktsiyada hisoblashimiz so'raladi va shuning uchun biz ushbu bo'limda ko'rganimizdek, bu chegara mavjudligini (va agar mavjud bo'lsa, uning qiymatini) aniqlash uchun ikkita bir tomonlama chegaralarni ko'rib chiqishimiz kerak.

\[\mathop {\lim }\limits_{t \to \, - {1^{\, - }}} \frac{{t + 1}}{{\left| {t + 1} \right|}} = \mathop {\lim }\limits_{t \to \, - {1^{\, - }}} \frac{{t + 1}}{{ - \left( {t + 1} \right)}} = \mathop {\lim }\limits_{t \to \, - {1^{\, - }}} - 1 = - 1\hspace{0.25in}{\mbox{recall }}t \to - {1^ - }{\mbox{ implies }}t < - 1\] \[\mathop {\lim }\limits_{t \to \, - {1^{\, + }}} \frac{{t + 1}}{{\left| {t + 1} \right|}} = \mathop {\lim }\limits_{t \to \, - {1^{\, + }}} \frac{{t + 1}}{{t + 1}} = \mathop {\lim }\limits_{t \to \, - {1^{\, + }}} 1 = 1\hspace{0.25in}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\mbox{recall }}t \to - {1^ + }{\mbox{ implies }}t > - 1\]

Shunday qilib, bu muammo uchun biz buni ko'rishimiz mumkin,

\[\mathop {\lim }\limits_{t \to \, - {1^{\, - }}} \frac{{t + 1}}{{\left| {t + 1} \right|}} = - 1 \ne 1 = \mathop {\lim }\limits_{t \to \, - {1^{\, + }}} \frac{{t + 1}}{{\left| {t + 1} \right|}}\]

va shuning uchun umumiy chegara mavjud emas.