Calculus I - 2.2-bo'lim: Limit tushunchasi (4-masala yechimi) | mathmodel.uz

2.2-bo'lim : Limit

4. Quyida $f\left( x \right)$ grafigi keltirilgan. Berilgan nuqtalarning har biri uchun $f\left( a \right)$ va $\mathop {\lim }\limits_{x \to a} f\left( x \right)$ qiymatini aniqlang. Agar biron bir miqdor mavjud bo'lmasa, sababini aniq tushuntiring.

$a = - 3$
$a = - 1$
$a = 2$
$a = 4$

$The graph of a function on $-5 \le x \le 5$ made up of three pieces. The first piece starts at (-5,1) and rises to a closed dot at (-3,4). The second piece starts at an open dot at (-3,-2), rises through an open dot at (-1,1) to a peak of about 1.7 near $x = 0.5$ and then falls to an open dot at (2,1). There is also a closed dot at (-1,3) above the open dot at $x = -1$. The third piece starts at the same open dot at (2,1), rises steeply to a peak at a closed dot at (4,5) and then falls to (5,3).$

a $a = - 3$ Yechim

Grafikdan shuni ko'rishimiz mumkinki,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( { - 3} \right) = 4}}\]

chunki yopiq nuqta $y = 4$ qiymatida.

Shuningdek, biz $x = - 3$ ga har ikki tomondan yaqinlashganimizda, grafik turli qiymatlarga yaqinlashayotganini ko'rishimiz mumkin (chapdan 4 va o'ngdan -2). Shu tufayli biz olamiz,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to - 3} f\left( x \right)\,\,\,{\mbox{does not exist}}}}\]

Har doim esda tutingki, chegaraning qiymati aslida funktsiyaning ko'rib chiqilayotgan nuqtadagi qiymatiga bog'liq emas. Limitning qiymati faqat ko'rib chiqilayotgan nuqta atrofidagi funktsiya qiymatlariga bog'liq. Ko'pincha ikkalasi boshqacha bo'ladi.

b $a = - 1$ Yechim

Grafikdan shuni ko'rishimiz mumkinki,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( { - 1} \right) = 3}}\]

chunki yopiq nuqta $y = 3$ qiymatida.

Bundan tashqari, biz $x = - 1$ ga har ikki tomondan yaqinlashganimizda, grafik bir xil qiymatga, 1 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:

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

c $a = 2$ Yechim

$x = 2$ uchun yopiq nuqta yo'qligi sababli biz buni ko'rishimiz mumkin,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( 2 \right)\,\,\,{\mbox{does not exist}}}}\]

Bundan tashqari, biz $x = 2$ ga har ikki tomondan yaqinlashganimizda, grafik bir xil qiymatga, 1 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:

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

d $a = 4$ Yechim

Grafikdan shuni ko'rishimiz mumkinki,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( 4 \right) = 5}}\]

chunki yopiq nuqta $y = 5$ qiymatida.

Bundan tashqari, biz $x = 4$ ga har ikki tomondan yaqinlashganimizda, grafik bir xil qiymatga, 5 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to 4} f\left( x \right) = 5}}\]