Функцияның шегінің қасиеттері және оларға арифметикалық амалдарды пайдалану.
Арифметикалық амалдарды дәлелдеусіз қысқаша келтірелік. , делік. Сонда
10. ![](data:image/png;base64,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) ![](data:image/png;base64,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) .
20. ![](data:image/png;base64,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) ![](data:image/png;base64,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) .
30. ![](data:image/png;base64,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) , .
, функциялары үшін болғанда (27 п.) «анықталмаған өрнектер» болады.
Егерде , жиынында монотонды функция болса, яғни монотонды өспелі (кемімейтін), монотонды кемімелі (өспейтін) функция болғанда (28 п. 1.5) теоремасы орындалады. 1.5 теореманы функциясы үшін былай жазуға болады.
Теорема. Егерде , функциясы монотонды өспелі болып, жоғарыдан шенелген болса, , онда болғанда функциясының шектелген шегі болады. Ал жоғарыдан шенелмеген болса, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAAICAIAAABoNngJAAABD0lEQVR4nGP5//8/A/UACxAvWrSooaHh9u3bt27diouLO336NKa6O3fuqKiooAl++PAhNDT0yJEj2dnZPT09UOOARuzbt6+jo+PNmze7du3Cau2XL1/27Nnj4uKCLFhRUWFnZ+fr68vIyIhwHRDU1taamZkBnSAoKAgRERMTA5qOrNnc3BzNuE2bNl25cqW8vDwqKgph3NOnT6dOndrW1paXl7d48WKIxKtXr/4hgfPnz2OG8u/fv9+/f3/w4EGgXoRxbm5uu3fvBjqnq6sLaC4wICByTGAAd6yysjKacUVFRU1NTS9evFi4cGFJSQnUuKtXr0Kk7969izXggADTLCCorKwEkkCz4CIsuPSTBwAdsX9Q5fWpEgAAAABJRU5ErkJggg==) .
Егерде жоғарыдан шенелмеген болса, онда алдын-ала алынған үлкен саны үшін жиынынан -ты табуға болады да, орындалады. -тан ![](data:image/png;base64,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) . Бұл деген сөз. Дәлелденді.
Теорема. Егерде , функциясы монотонды кемімелі болып, төменнен шенелген болса ( ), онда болғанда функциясының шектелген шегі болады. Ал, төменнен шенелмеген болса, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABoAAAAICAIAAABoNngJAAABD0lEQVR4nGP5//8/A/UACxAvWrSooaHh9u3bt27diouLO336NKa6O3fuqKiooAl++PAhNDT0yJEj2dnZPT09UOOARuzbt6+jo+PNmze7du3Cau2XL1/27Nnj4uKCLFhRUWFnZ+fr68vIyIhwHRDU1taamZkBnSAoKAgRERMTA5qOrNnc3BzNuE2bNl25cqW8vDwqKgph3NOnT6dOndrW1paXl7d48WKIxKtXr/4hgfPnz2OG8u/fv9+/f3/w4EGgXoRxbm5uu3fvBjqnq6sLaC4wICByTGAAd6yysjKacUVFRU1NTS9evFi4cGFJSQnUuKtXr0Kk7969izXggADTLCCorKwEkkCz4CIsuPSTBwAdsX9Q5fWpEgAAAABJRU5ErkJggg==) .
Достарыңызбен бөлісу: |