2.3-bo'lim : Bir tomonlama chegaralar
1. Quyida \(f\left( x \right)\) grafigi keltirilgan. Berilgan nuqtalarning har biri uchun \(f\left( a \right)\), \(\mathop {\lim }\limits_{x \to {a^{\, - }}} f\left( x \right)\), \(\mathop {\lim }\limits_{x \to {a^{\, + }}} f\left( x \right)\) va \(\mathop {\lim }\limits_{x \to a} f\left( x \right)\) qiymatini aniqlang. Agar biron bir miqdor mavjud bo'lmasa, sababini aniq tushuntiring.
- \(a = - 4\)
- \(a = - 1\)
- \(a = 2\)
- \(a = 4\)
a \(a = - 4\) Yechim
Grafikdan shuni ko'rishimiz mumkinki,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( { - 4} \right) = 3}}\]chunki yopiq nuqta \(y = 3\) qiymatida.
Shuningdek, biz \(x = - 4\) ga chapdan yaqinlashganimizda grafik 3 qiymatiga, o‘ngdan yaqinlashganda esa -2 qiymatiga yaqinlashayotganini ko‘rishimiz mumkin. Shunday qilib, biz olamiz,
\[\require{bbox} \bbox[4pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \, - {4^{\, - }}} f\left( x \right) = 3\hspace{0.25in}\hspace{0.25in}\& \hspace{0.25in}\mathop {\lim }\limits_{x \to \, - {4^{\, + }}} f\left( x \right) = - 2}}\]Endi, ikkita bir tomonlama chegaralar har xil bo'lgani uchun biz buni bilamiz,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \, - 4} f\left( x \right)\,\,{\mbox{does not exist}}}}\]b \(a = - 1\) Yechim
Grafikdan shuni ko'rishimiz mumkinki,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( { - 1} \right) = 4}}\]chunki yopiq nuqta \(y = 4\) qiymatida.
Bundan tashqari, biz \(x = - 1\) ga har ikki tomondan yaqinlashganimizda, grafik bir xil qiymatga, 4 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:
\[\require{bbox} \bbox[4pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \, - {1^{\, - }}} f\left( x \right) = 4\hspace{0.25in}\hspace{0.25in}\& \hspace{0.25in}\mathop {\lim }\limits_{x \to \, - {1^{\, + }}} f\left( x \right) = 4}}\]Ikki bir tomonlama chegaralar bir xil, shuning uchun biz bilamiz,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \, - 1} f\left( x \right) = 4}}\]c \(a = 2\) Yechim
Grafikdan shuni ko'rishimiz mumkinki,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( 2 \right) = - 1}}\]chunki yopiq nuqta \(y = - 1\) qiymatida.
Shuningdek, biz \(x = 2\) ga chapdan yaqinlashganimizda, grafik -1 qiymatiga yaqinlashayotganini va o'ngdan yaqinlashganda grafik 5 qiymatiga yaqinlashayotganini ko'rishimiz mumkin.
\[\require{bbox} \bbox[4pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \,{2^{\, - }}} f\left( x \right) = - 1\hspace{0.25in}\hspace{0.25in}\& \hspace{0.25in}\mathop {\lim }\limits_{x \to \,{2^{\, + }}} f\left( x \right) = 5}}\]Endi, ikkita bir tomonlama chegaralar har xil bo'lgani uchun biz buni bilamiz,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \,2} f\left( x \right)\,\,{\mbox{does not exist}}}}\]d \(a = 4\) Yechim
\(x = 4\) uchun yopiq nuqta yo'qligi sababli biz buni ko'rishimiz mumkin,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( 4 \right)\,\,\,{\mbox{does not exist}}}}\]Bundan tashqari, biz \(x = 4\) ga har ikki tomondan yaqinlashganimizda, grafik bir xil qiymatga, 2 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:
\[\require{bbox} \bbox[4pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \,{4^{\, - }}} f\left( x \right) = 2\hspace{0.25in}\hspace{0.25in}\& \hspace{0.25in}\mathop {\lim }\limits_{x \to \,{4^{\, + }}} f\left( x \right) = 2}}\]Ikki bir tomonlama chegaralar bir xil, shuning uchun biz bilamiz,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \,4} f\left( x \right) = 2}}\]Har doim esda tutingki, chegaraning qiymati (shu jumladan bir tomonlama chegaralar) aslida funktsiyaning ushbu nuqtadagi qiymatiga bog'liq emas. Limitning qiymati faqat ko'rib chiqilayotgan nuqta atrofidagi funktsiya qiymatlariga bog'liq. Shuning uchun, bu nuqtada funktsiya mavjud bo'lmasa ham, chegara va bir tomonlama chegaralar hali ham qiymatga ega bo'lishi mumkin.