Las propiedades de los limites son procedimientos básicos que facilitan la resolución de los ejercicios. A continuación te presentamos dichas propiedades con algunos ejemplos de su aplicación.
Propiedades de los límites
Dentro de las propiedades de los límites tenemos:
1.- El límite de una función si existe es único.
2.- Limite de una función
![\[\displaystyle \lim_{x \to a}x=a\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-495b8102166d8f37d08eada30acda037_l3.png "Rendered by QuickLaTeX.com")
Ejemplo:
![\[\displaystyle \lim_{x \to 3}x=3\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-db8c21f85d513b1fa074519645d56be1_l3.png "Rendered by QuickLaTeX.com")
3.- Limite de una constante
![\[\displaystyle \lim_{x \to a}c=c\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-1510c49c6d26d6e9473eb8d6e3d060d3_l3.png "Rendered by QuickLaTeX.com")
Ejemplo:
![\[\displaystyle \lim_{x \to 2}8=8\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-bc26def751d9d84c354aed604d536cd0_l3.png "Rendered by QuickLaTeX.com")
4.- Limite de la suma o resta
![\[\displaystyle \lim_{x \to a}(F(x)+G(x))=\displaystyle \lim_{x \to a}F(x)+\displaystyle \lim_{x \to a}G(x)\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-2fa8dba66453dc2d955febb608fa6543_l3.png "Rendered by QuickLaTeX.com")
![\[\displaystyle \lim_{x \to a}(F(x)-G(x))=\displaystyle \lim_{x \to a}F(x)-\displaystyle \lim_{x \to a}G(x)\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-3755b383ad581e99de566c397f5efb1c_l3.png "Rendered by QuickLaTeX.com")
Ejemplo:
![\[\displaystyle \lim_{x \to 5}(x+x^{3}-x^{4})=\displaystyle \lim_{x \to 5}x+\displaystyle \lim_{x \to 5}x^{3}-\displaystyle \lim_{x \to 5}x^{4}\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-2e93381d034f4915291f7de03f00cec3_l3.png "Rendered by QuickLaTeX.com")
5.- Limite de un producto
![\[\displaystyle \lim_{x \to a}(F(x).G(x))=\displaystyle \lim_{x \to a}F(x).\displaystyle \lim_{x \to a}G(x)\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-af94f3623dac936f2a2303532218e9af_l3.png "Rendered by QuickLaTeX.com")
Ejemplo:
![\[\displaystyle \lim_{x \to 8}((x+1).(x+2))=\displaystyle \lim_{x \to 8}(x+1).\displaystyle \lim_{x \to 8}(x+2)\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-73840e00159fa7d60ffeee27ad4e974b_l3.png "Rendered by QuickLaTeX.com")
6.- Limite de una división
![\[\displaystyle \lim_{x \to a}(\frac{F(x)}{G(x)})=\frac{\displaystyle \lim_{x \to a}F(x)}{\displaystyle \lim_{x \to a}G(x)}\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-6e3b8a600e5f071c56f2045eb8ed48db_l3.png "Rendered by QuickLaTeX.com")
Ejemplo:
![\[\displaystyle \lim_{x \to 1}(\frac{x+1}{x})=\frac{\displaystyle \lim_{x \to 1}(x+1)}{\displaystyle \lim_{x \to 1}x}\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-08de107ad14d91c2f99cd6287677f814_l3.png "Rendered by QuickLaTeX.com")
7.-Limite de una potencia
![\[\displaystyle \lim_{x \to a}x^{n}= a^{n}\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-a63534441bba87d7d9000d146b8a8fc8_l3.png "Rendered by QuickLaTeX.com")
Ejemplo:
![\[\displaystyle \lim_{x \to 2}x^{3}= (2)^{3}\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-d125aa26c97315f923e22134791ca07b_l3.png "Rendered by QuickLaTeX.com")
8.- Limite de un escalar por una función
![\[\displaystyle \lim_{x \to a}k.F(x)= k\displaystyle \lim_{x \to a}F(x)\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-79bb30861b8bac3d7ea4a832bbbb40e8_l3.png "Rendered by QuickLaTeX.com")
donde K es el escalar
Ejemplo:
![\[\displaystyle \lim_{x \to 4}2.x= 2\displaystyle \lim_{x \to 4}x\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-b16da17671c94897722967907042627a_l3.png "Rendered by QuickLaTeX.com")
9.- Limite de la raíz enésima de una función
![\[\displaystyle \lim_{x \to a}\sqrt[n]{F(x)}=\sqrt[n]{\displaystyle \lim_{x \to a}F(x)}\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-fe9872bdbb85d82b27665ea5df2e4eb3_l3.png "Rendered by QuickLaTeX.com")
Ejemplo:
![\[\displaystyle \lim_{x \to 3}\sqrt[4]{x}=\sqrt[4]{\displaystyle \lim_{x \to 3}x}\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-4a59b6d8f3c8b0d855d662d9804d7ad8_l3.png "Rendered by QuickLaTeX.com")
10.- Límite de un logaritmo de una función
![\[\displaystyle \lim_{x \to a}(log_{b}F(x))=log_{b}(\displaystyle \lim_{x \to a}F(x))\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-a6b388486814195433894ffa1c1bea1d_l3.png "Rendered by QuickLaTeX.com")
![\[\displaystyle \lim_{x \to a}(LnF(x))=Ln(\displaystyle \lim_{x \to a}F(x))\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-199f938ff06e2cb0195f58dc53c3ed36_l3.png "Rendered by QuickLaTeX.com")
Ejemplo:
![\[\displaystyle \lim_{x \to 5}(log_{b}x)=log_{b}(\displaystyle \lim_{x \to 5}x)\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-6f47c6d602fc0dd6eb4a6e31f82cddf4_l3.png "Rendered by QuickLaTeX.com")
![\[\displaystyle \lim_{x \to 6}(Ln(x))=Ln(\displaystyle \lim_{x \to 6}(x))\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-5caa5bac0f0794824232496a97108c31_l3.png "Rendered by QuickLaTeX.com")
11.- Limite de una función elevada a otra función
![\[\displaystyle \lim_{x \to a}(F(x))^{G(x)} =\displaystyle \lim_{x \to a}(F(x))^{\displaystyle \lim_{x \to a}G(x)}\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-c04a233568f8ea72f9efe656641f6c8b_l3.png "Rendered by QuickLaTeX.com")
Ejemplo:
![\[\displaystyle \lim_{x \to 4}(x)^{x^{3}} =\displaystyle \lim_{x \to 4}(x)^{\displaystyle \lim_{x \to 4}x^{3}}\]](https://calculodiferencial.com/wp-content/ql-cache/quicklatex.com-c036cb2fb00b5c90ba27811658b6d746_l3.png "Rendered by QuickLaTeX.com")
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