2.4-bo'lim : Limit xususiyatlari
9. Quyidagi chegarani hisoblash uchun ushbu bo'limda berilgan limit xususiyatlaridan foydalaning. Har bir bosqichda foydalanilayotgan mulkni aniq ko'rsating. Agar biron bir chegarani hisoblashning iloji bo'lmasa, nima uchun buni aniq tushuntiring.
\[\mathop {\lim }\limits_{x \to 10} \left( {4x + \sqrt[3]{{x - 2}}} \right)\]
Maslahat: Cheklovni hisoblash uchun ushbu bo'limdagi 7, 8 va/yoki 9-xususiyatlardan foydalanmagunimizcha, limitning chegara xususiyatlaridan foydalanishimiz kerak.
Yechim
\[\begin{alignat*}{3}\mathop {\lim }\limits_{x \to 10} \left( {4x + \sqrt[3]{{x - 2}}} \right) & = \mathop {\lim }\limits_{x \to 10} 4x + \mathop {\lim }\limits_{x \to 10} \sqrt[3]{{x - 2}} & & \hspace{0.25in}{\mbox{Property 2}}\\ & = \mathop {\lim }\limits_{x \to 10} 4x + \sqrt[3]{{\mathop {\lim }\limits_{x \to 10} \left( {x - 2} \right)}} & & \hspace{0.25in}{\mbox{Property 6}}\\ & = \mathop {\lim }\limits_{x \to 10} 4x + \sqrt[3]{{\mathop {\lim }\limits_{x \to 10} x - \mathop {\lim }\limits_{x \to 10} 2}} & & \hspace{0.25in}{\mbox{Property 2}}\\ & = 4\mathop {\lim }\limits_{x \to 10} x + \sqrt[3]{{\mathop {\lim }\limits_{x \to 10} x - \mathop {\lim }\limits_{x \to 10} 2}} & & \hspace{0.25in}{\mbox{Property 1}}\\ & = 4\left( {10} \right) + \sqrt[3]{{10 - 2}} & & \hspace{0.25in}{\mbox{Properties 7 \& 8}}\\ & = 42 & & & \end{alignat*}\]