Rules for limits (function)
Let
denote a subset and
a point. Let
and
denote
functions,
such that the
limits
and
exist. Then the following statements hold.
- The sum
has in
the limit
-
![{\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,(f(x)+g(x))=\operatorname {lim} _{x\rightarrow a}\,f(x)+\operatorname {lim} _{x\rightarrow a}\,g(x)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70d14a343ba2ae5b6990acce77d349b5c684ee8a)
- The product
has in
the limit
-
![{\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,(f(x)\cdot g(x))=\operatorname {lim} _{x\rightarrow a}\,f(x)\cdot \operatorname {lim} _{x\rightarrow a}\,g(x)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b209832e18d982d7250763721fc161c4e0238f7)
- Suppose that
for all
and
.
Then the quotient
has in
the limit
-
![{\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,{\frac {f(x)}{g(x)}}={\frac {\operatorname {lim} _{x\rightarrow a}\,f(x)}{\operatorname {lim} _{x\rightarrow a}\,g(x)}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88418ff630e031b955a96d007de4da474b42d330)