2.4-bo'lim : Limit xususiyatlari

1. \(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = 5\) berilgan bo'lsa, \(\mathop {\lim }\limits_{x \to 0} g\left( x \right) = - 1\) va \(\mathop {\lim }\limits_{x \to 0} h\left( x \right) = - 3\) 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.
   1. \(\mathop {\lim }\limits_{x \to 0} \left[ {11 + 7f\left( x \right)} \right]\)
   2. \(\mathop {\lim }\limits_{x \to 0} \left[ {6 - 4g\left( x \right) - 10h\left( x \right)} \right]\)
   3. \(\mathop {\lim }\limits_{x \to 0} \left[ {4g\left( x \right) - 12f\left( x \right) + 3h\left( x \right)} \right]\)
   4. \(\mathop {\lim }\limits_{x \to 0} \left[ {g\left( x \right)\left( {1 + 2f\left( x \right)} \right)} \right]\)
2. \(\mathop {\lim }\limits_{x \to 12} f\left( x \right) = 2\) berilgan bo'lsa, \(\mathop {\lim }\limits_{x \to 12} g\left( x \right) = 6\) va \(\mathop {\lim }\limits_{x \to 12} h\left( x \right) = 9\) 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.
   1. \(\displaystyle \mathop {\lim }\limits_{x \to 12} \left[ {h\left( x \right)f\left( x \right) + \frac{{1 + g\left( x \right)}}{{g\left( x \right)}}} \right]\)
   2. \(\mathop {\lim }\limits_{x \to 12} \left[ {\left( {3 - f\left( x \right)} \right)\left( {1 + 2g\left( x \right)} \right)} \right]\)
   3. \(\displaystyle \mathop {\lim }\limits_{x \to 12} \frac{{f\left( x \right) + 1}}{{3g\left( x \right) - 2h\left( x \right)}}\)
   4. \(\displaystyle \mathop {\lim }\limits_{x \to 12} \frac{{f\left( x \right) - 2g\left( x \right)}}{{7 + h\left( x \right)f\left( x \right)}}\)
3. \(\mathop {\lim }\limits_{x \to \, - 1} f\left( x \right) = 0\) berilgan, \(\mathop {\lim }\limits_{x \to \, - 1} g\left( x \right) = 9\) va \(\mathop {\lim }\limits_{x \to \, - 1} h\left( x \right) = - 7\) 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.
   1. \(\mathop {\lim }\limits_{x \to \, - 1} \left[ {{{\left( {g\left( x \right)} \right)}^2} - {{\left( {h\left( x \right)} \right)}^3}} \right]\)
   2. \(\mathop {\lim }\limits_{x \to \, - 1} \sqrt {3 + 6f\left( x \right) - h\left( x \right)} \)
   3. \(\mathop {\lim }\limits_{x \to \, - 1} \sqrt {f\left( x \right) - g\left( x \right)h\left( x \right)} \)
   4. \(\displaystyle \mathop {\lim }\limits_{x \to \, - 1} \sqrt[4]{{\frac{{2 + g\left( x \right)}}{{1 - 10h\left( x \right)}}}}\)

Quyidagi chegaralarning har biri uchun limitni hisoblash uchun ushbu bo'limda berilgan chegara xususiyatlaridan foydalaning. Har bir bosqichda foydalanilayotgan mulkni aniq ko'rsating. Agar biron bir chegarani hisoblashning iloji bo'lmasa, nima uchun buni aniq tushuntiring.

1. \(\mathop {\lim }\limits_{x \to 4} \left( {3{x^2} - 9x + 2} \right)\)
2. \(\mathop {\lim }\limits_{w \to \, - 1} \left( {w - {{\left( {{w^2} + 3} \right)}^2}} \right)\)
3. \(\mathop {\lim }\limits_{t \to 0} \left( {{t^4} - 4{t^2} + 12t - 8} \right)\)
4. \(\displaystyle \mathop {\lim }\limits_{z \to 2} \frac{{10 + {z^2}}}{{3 - 4z}}\)
5. \(\displaystyle \mathop {\lim }\limits_{x \to \,7} \frac{{8x}}{{{x^2} - 14x + 49}}\)
6. \(\displaystyle \mathop {\lim }\limits_{y \to \, - 3} \frac{{{y^3} - 20y + 4}}{{{y^2} + 8y - 1}}\)
7. \(\mathop {\lim }\limits_{w \to \, - 6} \sqrt[3]{{8 + 7w}}\)
8. \(\mathop {\lim }\limits_{t \to 1} \left( {4{t^2} - \sqrt {8t + 1} } \right)\)
9. \(\mathop {\lim }\limits_{x \to 8} \left( {\sqrt[4]{{3x - 8}} + \sqrt {9 + 2x} } \right)\)