Epsiclas
Definiciones de límites de funciones
Matemáticas II> Análisis
 

Límites en el infinito

Finitos

\(\displaystyle\lim_{x \to \infty}{f(x)}=L\ si\ \forall \epsilon>0\ \exists\ x0\ /\ si\ x>x0\ \Rightarrow\ |f(x)-L|<\epsilon\)

\(\displaystyle\lim_{x \to -\infty}{f(x)}=L\ si\ \forall \epsilon>0\ \exists\ x0\ /\ si\ x<x0\ \Rightarrow\ |f(x)-L|<\epsilon\)

Infinitos

\(\displaystyle\lim_{x \to \infty}{f(x)}=\infty\ si\ \forall k\ \exists\ x0\ /\ si\ x>x0\ \Rightarrow\ f(x)>k\)

\(\displaystyle\lim_{x \to -\infty}{f(x)}=\infty\ si\ \forall k\ \exists\ x0\ /\ si\ x<x0\ \Rightarrow\ f(x)>k\)

\(\displaystyle\lim_{x \to \infty}{f(x)}=-\infty\ si\ \forall k\ \exists\ x0\ /\ si\ x>x0\ \Rightarrow\ f(x)<k\)

\(\displaystyle\lim_{x \to -\infty}{f(x)}=-\infty\ si\ \forall k\ \exists\ x0\ /\ si\ x<x0\ \Rightarrow\ f(x)<k\)

Límites puntuales

Finitos

\(\displaystyle\lim_{x \to c}{f(x)}=L\ si\ \forall \epsilon>0\ \exists\ \delta>0 \ /\ si\ |c-x|<\delta \ \Rightarrow\ |f(x)-L|<\epsilon\)

\(\displaystyle\lim_{x \to c^+}{f(x)}=L\ si\ \forall \epsilon>0\ \exists\ \delta>0 \ /\ si\ 0<c-x<\delta \ \Rightarrow\ |f(x)-L|<\epsilon\)

\(\displaystyle\lim_{x \to c^-}{f(x)}=L\ si\ \forall \epsilon>0\ \exists\ \delta>0 \ /\ si\ 0<x-c<\delta \ \Rightarrow\ |f(x)-L|<\epsilon\)

Infinitos

\(\displaystyle\lim_{x \to c}{f(x)}=\infty\ si\ \forall k>0\ \exists\ \delta>0 \ /\ si\ |c-x|<\delta \ \Rightarrow\ f(x)>k\)

\(\displaystyle\lim_{x \to c^+}{f(x)}=\infty\ si\ \forall k\ \exists\ \delta>0 \ /\ si\ 0<c-x<\delta \ \Rightarrow\ f(x)>k\)

\(\displaystyle\lim_{x \to c^-}{f(x)}=\infty\ si\ \forall k\ \exists\ \delta>0 \ /\ si\ 0<x-c<\delta \ \Rightarrow\ f(x)>k\)

\(\displaystyle\lim_{x \to c}{f(x)}=-\infty\ si\ \forall k\ \exists\ \delta>0 \ /\ si\ |c-x|<\delta \ \Rightarrow\ f(x)<k\)

\(\displaystyle\lim_{x \to c^+}{f(x)}=-\infty\ si\ \forall k\ \exists\ \delta>0 \ /\ si\ 0<c-x<\delta \ \Rightarrow\ f(x)<k\)

\(\displaystyle\lim_{x \to c^-}{f(x)}=-\infty\ si\ \forall k\ \exists\ \delta>0 \ /\ si\ 0<x-c<\delta \ \Rightarrow\ f(x)<k\)

 
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Epsiclas
Alberto Rodriguez Santos
Desde 11-11-2011
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