חשבון אינפיניטסימלי/גבולות: הבדלים בין גרסאות בדף

בסביבת <math>x=0</math>]]

<math>\lim _{x \to a\infty} \frac{{9x^2 + x - 4}}{{ - 2x^2 + 3x - 10}} = \lim _{x \to a\infty} \frac{{\frac{{9x^2 + x - 4}}{{x^2 }}}}{{\frac{{ - 2x^2 + 3x - 10}}{{x^2 }}}} = \lim _{x \to a\infty} \frac{{9 + \frac{1}{x} - \frac{4}{{x^2 }}}}{{ - 2 + \frac{3}{x} - \frac{{10}}{{x^2 }}}}</math>

<math>\lim _{x \to a\infty} \frac{{9 + \frac{1}{x} - \frac{4}{{x^2 }}}}{{ - 2 + \frac{3}{x} - \frac{{10}}{{x^2 }}}} = \frac{{\lim _{x \to a\infty} \left( {9 + \frac{1}{x} - \frac{4}{{x^2 }}} \right)}}{{\lim _{x \to a\infty} \left( { - 2 + \frac{3}{x} - \frac{{10}}{{x^2 }}} \right)}} = \frac{{\lim _{x \to a\infty} 9 + \lim _{x \to a\infty} \left( {\frac{1}{x}} \right) - \lim _{x \to a\infty} \left( {\frac{4}{{x^2 }}} \right)}}{{\lim _{x \to a\infty} \left( { - 2} \right) + \lim _{x \to a\infty} \left( {\frac{3}{x}} \right) - \lim _{x \to a\infty} \left( {\frac{{10}}{{x^2 }}} \right)}} =

<math> = \frac{{9 + 0 - 4\lim _{x \to a\infty} \left( {\frac{1}{{x^2 }}} \right)}}{{ - 2 + 3\lim _{x \to a\infty} \left( {\frac{1}{x}} \right) - 10\lim _{x \to a\infty} \left( {\frac{1}{{x^2 }}} \right)}} = \frac{{9 - 4 \cdot 0}}{{ - 2 + 3 \cdot 0 - 10 \cdot 0}} = \frac{9}{{ - 2}} = - 4.5</math>

<math> | x |=\left\{\begin{matrix} x & & x \ge 0 \\ -x & & x \ge< 0 \end{matrix}\right. </math>

<math>\lim _{x \to 0^ - } \left| -x \right| = \lim _{x \to 0^ - } (-x) = 0</math>