Asosiy tarkibga o'tish

12.7-bo'lim : Vektor funktsiyalari bilan hisoblash

3. Quyidagi limitni hisoblang.

\[\mathop {\lim }\limits_{t \to \infty } \left\langle {\frac{1}{{{t^2}}},\frac{{2{t^2}}}{{1 - t - {t^2}}},{{\bf{e}}^{ - t}}} \right\rangle \]
Yechim

Haqiqatan ham bu muammo bilan bu erda ko'p narsa qilish mumkin emas. Biz qilishimiz kerak bo'lgan narsa vektorning barcha komponentlarining chegarasini olishdir.

\[\begin{align*}\mathop {\lim }\limits_{t \to \infty } \left\langle {\frac{1}{{{t^2}}},\frac{{2{t^2}}}{{1 - t - {t^2}}},{{\bf{e}}^{ - t}}} \right\rangle & = \left\langle {\mathop {\lim }\limits_{t \to \infty } \frac{1}{{{t^2}}},\mathop {\lim }\limits_{t \to \infty } \frac{{2{t^2}}}{{1 - t - {t^2}}},\mathop {\lim }\limits_{t \to \infty } {{\bf{e}}^{ - t}}} \right\rangle \\ & = \left\langle {\mathop {\lim }\limits_{t \to \infty } \frac{1}{{{t^2}}},\mathop {\lim }\limits_{t \to \infty } \frac{{2{t^2}}}{{{t^2}\left( {\frac{1}{{{t^2}}} - \frac{1}{t} - 1} \right)}},\mathop {\lim }\limits_{t \to \infty } {{\bf{e}}^{ - t}}} \right\rangle = \require{bbox} \bbox[2pt,border:1px solid black]{{\left\langle {0, - 2,0} \right\rangle }}\end{align*}\]

Cheksiz jarayonlar/faktlardagi asosiy chegarangizni unutmang.