De Wikilibros, la colección de libros de texto de contenido libre.
Por definición,
f
(
x
)
=
O
(
g
(
x
)
)
{\displaystyle f(x)=O(g(x))}
x
→
∞
{\displaystyle x\to \infty }
lim
x
→
∞
f
(
x
)
g
(
x
)
<
∞
{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}<\infty }
Hallamos entonces el límite:
lim
n
→
∞
3
∗
(
n
+
1
)
7
+
2
n
log
n
n
7
=
{\displaystyle \lim _{n\to \infty }{\frac {3*(n+1)^{7}+2n\log n}{n^{7}}}=}
=
lim
n
→
∞
3
∗
(
n
+
1
)
7
+
2
n
log
n
n
7
{\displaystyle =\lim _{n\to \infty }{\frac {3*(n+1)^{7}+2n\log n}{n^{7}}}}
=
lim
n
→
∞
3
∗
(
n
+
1
)
7
n
7
+
lim
n
→
∞
2
n
log
n
n
7
{\displaystyle =\lim _{n\to \infty }{\frac {3*(n+1)^{7}}{n^{7}}}+\lim _{n\to \infty }{\frac {2n\log n}{n^{7}}}}
=
3
+
lim
n
→
∞
2
log
n
n
6
{\displaystyle =3+\lim _{n\to \infty }{\frac {2\log n}{n^{6}}}}
=
3
+
lim
n
→
∞
2
6
n
6
ln
10
=
3
{\displaystyle =3+\lim _{n\to \infty }{\frac {2}{6n^{6}\ln 10}}=3}
3
<
∞
⇒
{\displaystyle 3<\infty \Rightarrow }
f
(
x
)
=
O
(
g
(
x
)
)
{\displaystyle f(x)=O(g(x))}
x
→
∞
{\displaystyle x\to \infty }
