2.4-bo'lim : Limit xususiyatlari
1. Berilgan \(\mathop {\lim }\limits_{x \to 8} f\left( x \right) = - 9\), \(\mathop {\lim }\limits_{x \to 8} g\left( x \right) = 2\) va \(\mathop {\lim }\limits_{x \to 8} h\left( x \right) = 4\) quyidagi chegaralarning har birini hisoblash uchun ushbu bo'limda berilgan limit xususiyatlaridan foydalanadi. Agar biron bir chegarani hisoblashning iloji bo'lmasa, nima uchun buni aniq tushuntiring.
- \(\mathop {\lim }\limits_{x \to 8} \left[ {2f\left( x \right) - 12h\left( x \right)} \right]\)
- \(\mathop {\lim }\limits_{x \to 8} \left[ {3h\left( x \right) - 6} \right]\)
- \(\mathop {\lim }\limits_{x \to 8} \left[ {g\left( x \right)h\left( x \right) - f\left( x \right)} \right]\)
- \(\mathop {\lim }\limits_{x \to 8} \left[ {f\left( x \right) - g\left( x \right) + h\left( x \right)} \right]\)
a \(\mathop {\lim }\limits_{x \to 8} \left[ {2f\left( x \right) - 12h\left( x \right)} \right]\) Yechim
Mana bu chegara uchun ish. Har bir bosqichda foydalanilgan xususiyat (yoki xususiyatlar) ro'yxatga olinadi va ba'zi hollarda xususiyatlar ko'rsatilgan bosqichda bir necha marta ishlatilgan bo'lishi mumkinligini unutmang.
\[\begin{alignat*}{3}\mathop {\lim }\limits_{x \to 8} \left[ {2f\left( x \right) - 12h\left( x \right)} \right] & = \mathop {\lim }\limits_{x \to 8} \left[ {2f\left( x \right)} \right] - \mathop {\lim }\limits_{x \to 8} \left[ {12h\left( x \right)} \right] & & \hspace{0.25in}{\mbox{Property 2}} &\\ & = 2\mathop {\lim }\limits_{x \to 8} f\left( x \right) - 12\mathop {\lim }\limits_{x \to 8} h\left( x \right) & & \hspace{0.25in}{\mbox{Property 1}} & \\ & = 2\left( { - 9} \right) - 12\left( 4 \right)& & \hspace{0.25in}{\mbox{Plug in values of limits}} & \\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 66}} & & &\end{alignat*}\]b \(\mathop {\lim }\limits_{x \to 8} \left[ {3h\left( x \right) - 6} \right]\) Yechim
Mana bu chegara uchun ish. Har bir bosqichda foydalanilgan xususiyat (yoki xususiyatlar) ro'yxatga olinadi va ba'zi hollarda xususiyatlar ko'rsatilgan bosqichda bir necha marta ishlatilgan bo'lishi mumkinligini unutmang.
\[\begin{alignat*}{3}\mathop {\lim }\limits_{x \to 8} \left[ {3h\left( x \right) - 6} \right] & = \mathop {\lim }\limits_{x \to 8} \left[ {3h\left( x \right)} \right] - \mathop {\lim }\limits_{x \to 8} 6 & & \hspace{0.25in}{\mbox{Property 2}} & \\ & = 3\mathop {\lim }\limits_{x \to 8} h\left( x \right) - \mathop {\lim }\limits_{x \to 8} 6 & & \hspace{0.25in}{\mbox{Property 1}} & \\ & = 3\left( 4 \right) - 6 & & \hspace{0.25in}{\mbox{Plug in value of limits \& Property 7}} & \\ & = \require{bbox} \bbox[2pt,border:1px solid black]{6} & & & \end{alignat*}\]c \(\mathop {\lim }\limits_{x \to 8} \left[ {g\left( x \right)h\left( x \right) - f\left( x \right)} \right]\) Yechim
Mana bu chegara uchun ish. Har bir bosqichda foydalanilgan xususiyat (yoki xususiyatlar) ro'yxatga olinadi va ba'zi hollarda xususiyatlar ko'rsatilgan bosqichda bir necha marta ishlatilgan bo'lishi mumkinligini unutmang.
\[\begin{alignat*}{3}\mathop {\lim }\limits_{x \to 8} \left[ {g\left( x \right)h\left( x \right) - f\left( x \right)} \right] & = \mathop {\lim }\limits_{x \to 8} \left[ {g\left( x \right)h\left( x \right)} \right] - \mathop {\lim }\limits_{x \to 8} f\left( x \right) & & \hspace{0.25in}{\mbox{Property 2}}\\ & = \left[ {\mathop {\lim }\limits_{x \to 8} g\left( x \right)} \right]\left[ {\mathop {\lim }\limits_{x \to 8} h\left( x \right)} \right] - \mathop {\lim }\limits_{x \to 8} f\left( x \right) & & \hspace{0.25in}{\mbox{Property 3}}\\ & = \left( 2 \right)\left( 4 \right) - \left( { - 9} \right) & & \hspace{0.25in}{\mbox{Plug in values of limits}}\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{17}} & & & \end{alignat*}\]d \(\mathop {\lim }\limits_{x \to 8} \left[ {f\left( x \right) - g\left( x \right) + h\left( x \right)} \right]\) Yechim
Mana bu chegara uchun ish. Har bir bosqichda foydalanilgan xususiyat (yoki xususiyatlar) ro'yxatga olinadi va ba'zi hollarda xususiyatlar ko'rsatilgan bosqichda bir necha marta ishlatilgan bo'lishi mumkinligini unutmang.
\[\begin{alignat*}{3}\mathop {\lim }\limits_{x \to 8} \left[ {f\left( x \right) - g\left( x \right) + h\left( x \right)} \right] & = \mathop {\lim }\limits_{x \to 8} f\left( x \right) - \mathop {\lim }\limits_{x \to 8} g\left( x \right) + \mathop {\lim }\limits_{x \to 8} h\left( x \right) & & \hspace{0.25in}{\mbox{Property 2}}\\ & = - 9 - 2 + 4 & & \hspace{0.25in}{\mbox{Plug in values of limits}}\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 7}} & & & \end{alignat*}\]