2.3-bo'lim : Bir tomonlama chegaralar
2. 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.
- \(a = - 2\)
- \(a = 1\)
- \(a = 3\)
- \(a = 5\)
a \(a = - 2\) 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.
Bundan tashqari, biz \(x = - 2\) ga chapdan yaqinlashganimizda, grafik bir qiymatga emas, aksincha, vahshiy tebranishini va o'ngdan yaqinlashganda, grafik -1 qiymatiga yaqinlashayotganini ko'rishimiz mumkin. Shunday qilib, biz olamiz,
\[\require{bbox} \bbox[4pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \, - {2^{\, - }}} f\left( x \right){\mbox{ does not exist}}\hspace{0.25in} \& \hspace{0.25in},\mathop {\lim }\limits_{x \to \, - {2^{\, + }}} f\left( x \right) = - 1}}\]Eslatib o'tamiz, chegara mavjud bo'lishi uchun funktsiya bitta qiymatga yaqinlashishi kerak va shuning uchun bu holda, \(x = - 2\) ning chap tomonidagi grafik bitta qiymatga yaqinlashmaganligi sababli, chap chegara mavjud bo'lmaydi. Bu o'ng qo'l chegarasi mavjud bo'lmaydi degani emas. Bu holda \(x = - 2\) ning o'ng tomonidagi grafik bitta qiymatga yaqinlashsa, o'ng chegara mavjud bo'ladi.
Endi ikkita bir tomonlama chegaralar har xil bo'lgani uchun biz buni bilamiz,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \, - 2} f\left( x \right)\,\,{\mbox{does not exist}}}}\]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, 3 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:
\[\require{bbox} \bbox[4pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \,{1^{\, - }}} f\left( x \right) = 3\hspace{0.25in}\& \hspace{0.25in}\mathop {\lim }\limits_{x \to \,{1^{\, + }}} f\left( x \right) = 3}}\]Ikki bir tomonlama chegaralar bir xil, shuning uchun biz bilamiz,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \,1} f\left( x \right) = 3}}\]c \(a = 3\) Yechim
Grafikdan shuni ko'rishimiz mumkinki,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( 3 \right) = - 2}}\]chunki yopiq nuqta \(y = - 2\) qiymatida.
Shuningdek, biz \(x = 2\) ga chapdan yaqinlashganimizda grafik 1 qiymatiga, o‘ngdan yaqinlashganda esa -3 qiymatiga yaqinlashayotganini ko‘rishimiz mumkin. Shunday qilib, biz olamiz,
\[\require{bbox} \bbox[4pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \,{3^{\, - }}} f\left( x \right) = 1\hspace{0.25in}\& \hspace{0.25in}\mathop {\lim }\limits_{x \to \,{3^{\, + }}} f\left( x \right) = - 3}}\]Endi, ikkita bir tomonlama chegaralar har xil bo'lgani uchun biz buni bilamiz,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \,3} f\left( x \right)\,\,{\mbox{does not exist}}}}\]d \(a = 5\) Yechim
Grafikdan shuni ko'rishimiz mumkinki,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( 5 \right) = 4}}\]chunki yopiq nuqta \(y = 4\) qiymatida.
Bundan tashqari, biz \(x = 5\) ga har ikki tomondan yaqinlashganimizda, grafik bir xil qiymatga, 4 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:
\[\require{bbox} \bbox[4pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \,{5^{\, - }}} f\left( x \right) = 4\hspace{0.25in}\& \hspace{0.25in}\mathop {\lim }\limits_{x \to \,{5^{\, + }}} f\left( x \right) = 4}}\]Ikki bir tomonlama chegaralar bir xil, shuning uchun biz bilamiz,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to \,5} f\left( x \right) = 4}}\]