Calculus I - 2.2-bo'lim: Limit tushunchasi (6-masala yechimi) | mathmodel.uz

2.2-bo'lim : Limit

6. 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.

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

$The graph of a function on $-4 \le x \le 4$ with a vertical asymptote at $x = -2$. To the left of the asymptote the graph starts at about (-4,1.5) and increases toward positive infinity as it approaches $x = -2$. To the right of the asymptote the graph comes up from negative infinity and rises to an open dot at (-1,1). There is a closed dot at (-1,3) above the open dot at $x = -1$. From the open dot at (-1,1) the graph falls to its lowest point at an open dot at (1,-3), then rises through a closed dot at (3,4) and continues upward.$

a $a = - 2$ 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 yaqinlashganda, grafik ikkala tomondan qiymatga yaqinlashmasligini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:

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

b $a = - 1$ Yechim

Grafikdan shuni ko'rishimiz mumkinki,

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

chunki yopiq nuqta $y = 3$ qiymatida.

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

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

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.

c $a = 1$ Yechim

$x = 1$ uchun yopiq nuqta yo'qligi sababli biz buni ko'rishimiz mumkin,

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

Shuningdek, biz $x = 1$ ga har ikki tomondan yaqinlashganimizda, grafik bir xil qiymatga -3 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:

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

d $a = 3$ 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.

Bundan tashqari, biz $x = 3$ ga har ikki tomondan yaqinlashganimizda, grafik bir xil qiymatga, 4 ga yaqinlashayotganini ko'rishimiz mumkin va shuning uchun biz quyidagilarni olamiz:

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