Asosiy tarkibga o'tish

2.6-bo'lim : Cheksiz limitlar

6. \(\displaystyle R\left( y \right) = \tan \left( y \right)\) baholash uchun,

  1. \(\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, - }}} R\left( y \right)\)
  2. \(\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, + }}} R\left( y \right)\)
  3. \(\mathop {\lim }\limits_{y \to {\textstyle{{3\pi } \over 2}}} R\left( y \right)\)

a Yechim

Bu muammoni hal qilishning eng oson yo'li tangens funksiya grafigidir, shuning uchun bu erda bir necha davrlar uchun tangens funksiyasining qisqacha chizmasi keltirilgan.

The graph of \(R\left( y \right) = \tan \left( y \right)\) over several periods with dashed vertical asymptotes at \(y = - \frac{\pi }{2}\), \(\frac{\pi }{2}\), \(\frac{{3\pi }}{2}\) and \(\frac{{5\pi }}{2}\). Between each pair of asymptotes the graph is an increasing curve that comes up from negative infinity on the left, crosses the y-axis at the midpoint and rises to positive infinity on the right.

Eskizdan shuni ko'rishimiz mumkinki,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, - }}} \tan \left( y \right) = \infty }}\]

b Yechim

Birinchi qismdagi grafikdan shuni ko'rishimiz mumkinki,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, + }}} \tan \left( y \right) = - \infty }}\]

c Yechim

Birinchi ikki qismdan boshlab,

\[\mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, - }}} R\left( y \right) \ne \mathop {\lim }\limits_{y \to {{{\textstyle{{3\pi } \over 2}}}^{\, + }}} R\left( y \right)\]

Shunday qilib, bizning asosiy chegara xususiyatlaridan biz \(\mathop {\lim }\limits_{y \to {\textstyle{{3\pi } \over 2}}} R\left( y \right)\) ekanligini ko'rishimiz mumkin mavjud emas.