यहाँ कुछ प्रमुख एवं महत्वपूर्ण गणितीय फलनों की सीमाएँ (limit) दी गई हैं। *a* और *b* दोनों नियतांक हैं (*x* के सापेक्ष)।

सामान्य फलनों की सीमाएँ
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{\displaystyle {\text{If }}\lim \_{x\to c}f(x)=L\_{1}{\text{ and }}\lim \_{x\to c}g(x)=L\_{2}{\text{ then:}}}
{\displaystyle \lim \_{x\to c}\,[f(x)\pm g(x)]=L\_{1}\pm L\_{2}}
{\displaystyle \lim \_{x\to c}\,[f(x)g(x)]=L\_{1}\times L\_{2}}
{\displaystyle \lim \_{x\to c}{\frac {f(x)}{g(x)}}={\frac {L\_{1}}{L\_{2}}}\qquad {\text{ if }}L\_{2}\neq 0}
{\displaystyle \lim \_{x\to c}\,f(x)^{n}=L\_{1}^{n}\qquad {\text{ if }}n{\text{ is a positive integer}}}
{\displaystyle \lim \_{x\to c}\,f(x)^{1 \over n}=L\_{1}^{1 \over n}\qquad {\text{ if }}n{\text{ is a positive integer, and if }}n{\text{ is even, then }}L\_{1}>0}
{\displaystyle \lim \_{x\to c}{\frac {f(x)}{g(x)}}=\lim \_{x\to c}{\frac {f'(x)}{g'(x)}}\qquad {\text{ if }}\lim \_{x\to c}f(x)=\lim \_{x\to c}g(x)=0{\text{ or }}\lim \_{x\to c}g(x)=\pm \infty } (एल् हॉस्पिटल नियम L'Hôpital's rule)
{\displaystyle \lim \_{h\to 0}{f(x+h)-f(x) \over h}=f'(x)}
{\displaystyle \lim \_{h\to 0}\left({\frac {f(x+h)}{f(x)}}\right)^{\frac {1}{h}}=\exp \left({\frac {f'(x)}{f(x)}}\right)}
{\displaystyle \lim \_{h\to 0}{\left({f(x(1+h)) \over {f(x)}}\right)^{1 \over {h}}}=\exp \left({\frac {xf'(x)}{f(x)}}\right)}

उल्लेखनीय विशिष्ट सीमाएँ
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{\displaystyle \lim \_{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{mx}=e^{mk}}
{\displaystyle \lim \_{x\to +\infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}}
{\displaystyle \lim \_{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{x}=e^{k}}
{\displaystyle \lim \_{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e}
{\displaystyle \lim \_{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\text{...}}+{\sqrt {2}}}}}}}} \_{n}=\pi }
{\displaystyle \lim \_{x\to 0}\left({\frac {a^{x}-1}{x}}\right)=\log {a},\qquad \forall ~a>0}

सरल फलन
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{\displaystyle \lim \_{x\to c}a=a}
{\displaystyle \lim \_{x\to c}x=c}
{\displaystyle \lim \_{dux\to c}ax+b=ac+b}
{\displaystyle \lim \_{x\to c}x^{r}=c^{r}\qquad {\mbox{ if }}r{\mbox{ is a positive integer}}}
{\displaystyle \lim \_{x\to 0^{+}}{\frac {1}{x^{r}}}=+\infty }
{\displaystyle \lim \_{x\to 0^{-}}{\frac {1}{x^{r}}}={\begin{cases}-\infty ,&{\text{if }}r{\text{ is odd}}\\+\infty ,&{\text{if }}r{\text{ is even}}\end{cases}}}

लघुगणकीय तथा चरघातांकी फलन
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{\displaystyle \lim \_{x\to 1}{\frac {\ln(x)}{x-1}}=1}
{\displaystyle {\mbox{For }}a>1:\,}
{\displaystyle \lim \_{x\to 0^{+}}\log \_{a}x=-\infty }
{\displaystyle \lim \_{x\to \infty }\log \_{a}x=\infty }
{\displaystyle \lim \_{x\to -\infty }a^{x}=0}
{\displaystyle {\mbox{If }}a<1:\,}
{\displaystyle \lim \_{x\to -\infty }a^{x}=\infty }

त्रिकोणमितीय फलन
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{\displaystyle \lim \_{x\to a}\sin x=\sin a}
{\displaystyle \lim \_{x\to a}\cos x=\cos a}
यदि {\displaystyle x} रेडियन में हो तो:

{\displaystyle \lim \_{x\to 0}{\frac {\sin x}{x}}=1}
{\displaystyle \lim \_{x\to 0}{\frac {1-\cos x}{x}}=0}
{\displaystyle \lim \_{x\to 0}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}}
{\displaystyle \lim \_{x\to n^{\pm }}\tan \left(\pi x+{\frac {\pi }{2}}\right)=\mp \infty \qquad {\text{for any integer }}n}
{\displaystyle \lim \_{x\to 0}{\frac {\sin ax}{x}}=a}
{\displaystyle \lim \_{x\to 0}{\frac {\sin ax}{\sin bx}}={\frac {a}{b}}}

अनन्त के पास
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{\displaystyle \lim \_{x\to \infty }N/x=0{\text{ for any real }}N}
{\displaystyle \lim \_{x\to \infty }x/N={\begin{cases}\infty ,&N>0\\{\text{does not exist}},&N=0\\-\infty ,&N<0\end{cases}}}
{\displaystyle \lim \_{x\to \infty }x^{N}={\begin{cases}\infty ,&N>0\\1,&N=0\\0,&N<0\end{cases}}}
{\displaystyle \lim \_{x\to \infty }N^{x}={\begin{cases}\infty ,&N>1\\1,&N=1\\0,&0<N<1\end{cases}}}
{\displaystyle \lim \_{x\to \infty }N^{-x}=\lim \_{x\to \infty }1/N^{x}=0{\text{ for any }}N>1}
{\displaystyle \lim \_{x\to \infty }{\sqrt[{x}]{N}}={\begin{cases}1,&N>0\\0,&N=0\\{\text{does not exist}},&N<0\end{cases}}}
{\displaystyle \lim \_{x\to \infty }{\sqrt[{N}]{x}}=\infty {\text{ for any }}N>0}
{\displaystyle \lim \_{x\to \infty }\log x=\infty }
{\displaystyle \lim \_{x\to 0^{+}}\log x=-\infty }