តារាងលីមីត

ខាងក្រោមនេះជាតារាងលីមីតអនុគមន៍ទូទៅ។ កត់សំគាល់ថា a និងb ជាចំនួនថេរតាមអថេរ x ។

លីមីតនៃអនុគមន៍ទូទៅ

ប្រសិនបើ $\underset{x\to c}{\lim }f(x)=L_1$ និង $\underset{x\to c}{\lim }g(x)=L_2$ នោះគេបាន

$\underset{x\to c}{\lim }[f(x)\pm g(x)]=L_1\pm L_2$

$\underset{x\to c}{\lim }[f(x)g(x)]=L_1\times L_2$

$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\frac{L_1}{L_2}$ ប្រសិនបើ $L_2\ne 0$

$\underset{x\to c}{\lim }f(x{)}^n=L_1^n$ ប្រសិនបើ n ជាចំនួនគត់វិជ្ជមាន

$\underset{x\to c}{\lim }f(x{)}^{\frac{1}{n}}=L_1^{\frac{1}{n}}$ ប្រសិនបើ n ជាចំនួនគត់វិជ្ជមានគូ និង$L_1>0$

$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\underset{x\to c}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$ ប្រសិនបើ $\underset{x\to c}{\lim }f(x)=\underset{x\to c}{\lim }g(x)=0$ ឬ $\underset{x\to c}{\lim }|g(x)|=+\mathrm{\infty }$ (ច្បាប់ឡូពីតាល់)

លីមីតនៃអនុគមន៍សាមញ្ញ

$\underset{x\to c}{\lim }a=a$

$\underset{x\to c}{\lim }x=c$

$\underset{x\to c}{\lim }ax+b=ac+b$

$\underset{x\to c}{\lim }{x}^r=c^r$ ប្រសិនបើ r ជាចំនួនគត់វិជ្ជមាន

$\underset{x\to 0^{+}}{\lim }\frac{1}{x^r}=+\mathrm{\infty }$

$\underset{x\to 0^{-}}{\lim }\frac{1}{x^r}=\{\begin{array}{cc}-\mathrm{\infty },& \text{if }r\text{ is odd}\\ +\mathrm{\infty },& \text{if }r\text{ is even}\end{array}$

លីមីតនៃអនុគមន៍លោការីត និងអនុគមន៍អ៊ិចស្ប៉ូណង់ស្យែល

ចំពោះ $a>1:$

$\underset{x\to 0^{+}}{\lim }{\log }_{a}x=-\mathrm{\infty }$

$\underset{x\to \mathrm{\infty }}{\lim }{\log }_{a}x=\mathrm{\infty }$

$\underset{x\to -\mathrm{\infty }}{\lim }{a}^x=0$

$\underset{x\to \mathrm{\infty }}{\lim }{a}^x=\mathrm{\infty }$

លីមីតនៃអនុគមន៍ត្រីកោណមាត្រ

$\underset{x\to a}{\lim }\sin x=\sin a$

$\underset{x\to a}{\lim }\cos x=\cos a$

$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$

$\underset{x\to 0}{\lim }\frac{1-\cos x}{x}=0$

$\underset{x\to 0}{\lim }\frac{1-\cos x}{x^2}=\frac{1}{2}$

$\underset{x\to n^{\pm }}{\lim }\tan (\pi x+\frac{\pi }{2})=\mp \mathrm{\infty }$ ចំពោះគ្រប់ចំនួនគត់ n

លីមីត x ខិតទៅរកអនន្ត

$\underset{x\to \mathrm{\infty }}{\lim }N/x=0$ ចំពោះគ្រប់ចំនួនពិត N

$\underset{x\to \mathrm{\infty }}{\lim }x/N=\{\begin{array}{cc}\mathrm{\infty },& N>0\\ \text{does not exist},& N=0\\ -\mathrm{\infty },& N<0\end{array}$

$\underset{x\to \mathrm{\infty }}{\lim }{x}^N=\{\begin{array}{cc}\mathrm{\infty },& N>0\\ 1,& N=0\\ 0,& N<0\end{array}$

$\underset{x\to \mathrm{\infty }}{\lim }{N}^x=\{\begin{array}{cc}\mathrm{\infty },& N>1\\ 1,& N=1\\ 0,& N<1\end{array}$

$\underset{x\to \mathrm{\infty }}{\lim }{N}^{-x}=\underset{x\to \mathrm{\infty }}{\lim }1/N^x=0\text{ for any }N>1$

$\underset{x\to \mathrm{\infty }}{\lim }\sqrt[x]{N}=\{\begin{array}{cc}1,& N>0\\ 0,& N=0\\ \text{does not exist},& N<0\end{array}$

$\underset{x\to \mathrm{\infty }}{\lim }\sqrt[N]{x}=\mathrm{\infty }$ ចំពោះ$N>0$

$\underset{x\to \mathrm{\infty }}{\lim }\log x=\mathrm{\infty }$

$\underset{x\to 0^{+}}{\lim }\log x=-\mathrm{\infty }$

ឧទាហរណ៍

$\underset{x\to 3}{\lim }{x}^2=9$

$\underset{x\to 0+}{\lim }{x}^x=1$

$\underset{x\to \mathrm{\infty }}{\lim }\frac{1}{x}=0$

$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$

$\underset{x\to 0}{\lim }\cos (cx{)}^{2/x^2}=e^{-c^2}$

$\underset{n\to \mathrm{\infty }}{\lim }{2}^{1/n}=\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{2}=1$

$\underset{n\to \mathrm{\infty }}{\lim }{n}^{1/n}=1$

$\underset{n\to \mathrm{\infty }}{\lim }\frac{2^n}{n!}=0$

$\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{x}{n})}^n=e^x$