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, \(x = 5\) da mutlaq qiymat ichidagi funktsiya nolga teng bo'lgani uchun biz buni ko'rishimiz mumkin,

\[\left| {x - 5} \right| = \left\{ {\begin{array}{*{20}{c}}{x - 5}&{x \ge 5}\\{ - \left( {x - 5} \right)}&{x < 5}\end{array}} \right.\]

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.

\[\mathop {\lim }\limits_{x \to {5^{\, - }}} \left( {10 + \left| {x - 5} \right|} \right) = \mathop {\lim }\limits_{x \to {5^{\, - }}} \left( {10 - \left( {x - 5} \right)} \right) = \mathop {\lim }\limits_{x \to {5^{\, - }}} \left( {15 - x} \right) = 10\hspace{0.25in}{\mbox{recall }}x \to {5^ - }{\mbox{ implies }}x < 5\] \[\mathop {\lim }\limits_{x \to {5^{\, + }}} \left( {10 + \left| {x - 5} \right|} \right) = \mathop {\lim }\limits_{x \to {5^{\, + }}} \left( {10 + \left( {x - 5} \right)} \right) = \mathop {\lim }\limits_{x \to {5^{\, + }}} \left( {5 + x} \right) = 10\hspace{0.25in}{\mbox{recall }}x \to {5^ + }{\mbox{ implies }}x > 5\]

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

\[\mathop {\lim }\limits_{x \to {5^{\, - }}} \left( {10 + \left| {x - 5} \right|} \right) = \mathop {\lim }\limits_{x \to 5 + } \left( {10 + \left| {x - 5} \right|} \right) = 10\]

va shuning uchun umumiy chegara mavjud bo'lishi kerak va,

\[\mathop {\lim }\limits_{x \to {5}} \left( {10 + \left| {x - 5} \right|} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{10}}\]