Calculus I - 2.2-bo'lim: Limit tushunchasi (5-masala yechimi) | mathmodel.uz

2.2-bo'lim : Limit

5. 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.

$a = - 8$
$a = - 2$
$a = 6$
$a = 10$

$The graph of a function on $-12 \le x \le 12$ made up of three pieces. The first piece starts at (-12,4), falls to its lowest point at an open dot at (-8,-6) and then rises to a closed dot at (-2,3). There is also a closed dot at (-8,-3) above the open dot at $x = -8$. The second piece comes down from positive infinity just to the right of $x = -2$, reaches a low point of about 1 near $x = 0$, rises to about 3 near $x = 2.5$, dips slightly and ends at an open dot at (6,2). The third piece starts at a closed dot at (6,5), rises to a peak of about 5.5 near $x = 7$ and then falls through a closed dot at (10,0) to about -4 at $x = 12$.$

a $a = - 8$ Yechim

Grafikdan shuni ko'rishimiz mumkinki,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( { - 8} \right) = - 3}}\]

chunki yopiq nuqta $y = - 3$ qiymatida.

Shuningdek, biz $x = - 8$ ga har ikki tomondan yaqinlashganimizda, grafik bir xil qiymatga -6 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to - 8} f\left( x \right) = - 6}}\]

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 $a = - 2$ Yechim

Grafikdan shuni ko'rishimiz mumkinki,

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

chunki yopiq nuqta $y = 3$ qiymatida.

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

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

c $a = 6$ Yechim

Grafikdan shuni ko'rishimiz mumkinki,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( 6 \right) = 5}}\]

chunki yopiq nuqta $y = 5$ qiymatida.

Shuningdek, biz $x = 6$ ga har ikki tomondan yaqinlashganda, grafik turli qiymatlarga yaqinlashayotganini ko'rishimiz mumkin (chapdan 2 va o'ngdan 5). Shu tufayli biz olamiz,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to 6} f\left( x \right)\,\,\,{\mbox{does not exist}}}}\]

d $a = 10$ Yechim

Grafikdan shuni ko'rishimiz mumkinki,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( {10} \right) = 0}}\]

chunki yopiq nuqta $y = 0$ qiymatida.

Bundan tashqari, biz $x = 10$ ga har ikki tomondan yaqinlashganimizda, grafik bir xil qiymatga, 0 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{x \to 10} f\left( x \right) = 0}}\]