4.10-bo'lim : L'Hospital qoidalari va noaniq shakllar
9. \(\mathop {\lim }\limits_{x \to {1^ + }} \left[ {\left( {x - 1} \right)\tan \left( {\displaystyle \frac{\pi }{2}x} \right)} \right]\) ni baholash uchun L’Hospital qoidasidan foydalaning.
Yechim
Bu erda e'tibor berish kerak bo'lgan birinchi narsa - bu L'Hospital qoidasiga ruxsat beruvchi shaklda emas. L'Hospital qoidasi faqat ma'lum bir ratsional funktsiyalar sinfida ishlaydi va bu ratsional funktsiya emasligi aniq.
Ammo e'tibor bering, u quyidagi noaniq shaklda,
\[{\mbox{as}}\,\,\,x \to {1^ + }\hspace{0.5in}\left( {x - 1} \right)\tan \left( {\frac{\pi }{2}x} \right) \to \left( 0 \right)\left( { - \infty } \right)\]va biz ushbu bo'lim uchun eslatmalarda muhokama qilganimizdek, biz har doim bunday noaniq shaklni L'Hospital qoidasini qo'llash imkonini beradigan oqilona ifodaga aylantirishimiz mumkin.
2-qadam
Haqiqiy savol shundaki, biz birinchi atamani yoki ikkinchi atamani maxrajga o'tkazamizmi? Bir qarashda, har ikkala atama ham maxrajda unchalik foydali emasdek tuyulishi mumkin. Xususan, agar biz tangensni maxrajga o'tkazsak, biz \({}^{1}/{}_{{\tan }}\) shaklida atamani farqlashimiz kerak bo'ladi. Buni farqlash unchalik qiziq emasdek tuyuladi va biz tugatganimizdan keyin tartibsizlikka tushishimiz mumkin.
Biroq, aynan shu atama biz tezda aniq bo'ladigan sabablarga ko'ra maxrajga o'tmoqchimiz.
\[\mathop {\lim }\limits_{x \to {1^ + }} \left[ {\left( {x - 1} \right)\tan \left( {\frac{\pi }{2}x} \right)} \right] = \mathop {\lim }\limits_{x \to {1^ + }} \frac{{x - 1}}{{{}^{1}/{}_{{\tan \left( {\frac{\pi }{2}x} \right)}}}} = \mathop {\lim }\limits_{x \to {1^ + }} \frac{{x - 1}}{{\cot \left( {\frac{\pi }{2}x} \right)}}\]3-qadam
Tangensni maxrajga o'tkazgandan so'ng biroz soddalashtirib, biz unchalik yomon ko'rinmaydigan narsaga erishdik. Shuningdek, biz ushbu muammoning qolgan qismi juda oddiy bo'lishini ko'ramiz.
Davom etishdan oldin shuni ham ta'kidlashimiz kerakki,
\[{\mbox{as}}\,\,\,x \to {1^ + }\hspace{0.5in}\frac{{x - 1}}{{\cot \left( {\frac{\pi }{2}x} \right)}} \to \frac{0}{0}\]va biz bu borada L'Hospital qoidasidan foydalanishimiz mumkin.
4-qadam
L'Hospital qoidasini qo'llash,
\[\mathop {\lim }\limits_{x \to {1^ + }} \left[ {\left( {x - 1} \right)\tan \left( {\frac{\pi }{2}x} \right)} \right] = \mathop {\lim }\limits_{x \to {1^ + }} \frac{{x - 1}}{{\cot \left( {\frac{\pi }{2}x} \right)}} = \mathop {\lim }\limits_{x \to {1^ + }} \frac{1}{{ - \frac{\pi }{2}{{\csc }^2}\left( {\frac{\pi }{2}x} \right)}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{2}{\pi }}}\]