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Equivalencias asintóticas (infinitésimos)
Casos continuos: función de x, con u→0
sin
u
∼
u
\sin u \sim u
sin
u
∼
u
sin
(
3
x
)
∼
3
x
\sin(3x) \sim 3x
sin
(
3
x
)
∼
3
x
tan
u
∼
u
\tan u \sim u
tan
u
∼
u
tan
(
x
2
)
∼
x
2
\tan(x^{2}) \sim x^{2}
tan
(
x
2
)
∼
x
2
arcsin
u
∼
u
\arcsin u \sim u
arcsin
u
∼
u
arcsin
(
2
x
)
∼
2
x
\arcsin(2x) \sim 2x
arcsin
(
2
x
)
∼
2
x
arctan
u
∼
u
\arctan u \sim u
arctan
u
∼
u
arctan
(
x
2
)
∼
x
2
\arctan\!\left(\tfrac{x}{2}\right) \sim \tfrac{x}{2}
arctan
(
2
x
)
∼
2
x
sinh
u
∼
u
\sinh u \sim u
sinh
u
∼
u
sinh
(
4
x
)
∼
4
x
\sinh(4x) \sim 4x
sinh
(
4
x
)
∼
4
x
1
−
cos
u
∼
u
2
2
1-\cos u \sim \dfrac{u^{2}}{2}
1
−
cos
u
∼
2
u
2
1
−
cos
(
2
x
)
∼
2
x
2
1-\cos(2x) \sim 2x^{2}
1
−
cos
(
2
x
)
∼
2
x
2
e
u
−
1
∼
u
e^{u}-1 \sim u
e
u
−
1
∼
u
e
3
x
−
1
∼
3
x
e^{3x}-1 \sim 3x
e
3
x
−
1
∼
3
x
a
u
−
1
∼
u
ln
a
(
a
>
0
)
a^{u}-1 \sim u\ln a \quad(a>0)
a
u
−
1
∼
u
ln
a
(
a
>
0
)
2
x
−
1
∼
x
ln
2
2^{x}-1 \sim x\ln 2
2
x
−
1
∼
x
ln
2
ln
(
1
+
u
)
∼
u
\ln(1+u) \sim u
ln
(
1
+
u
)
∼
u
ln
(
1
+
5
x
)
∼
5
x
\ln(1+5x) \sim 5x
ln
(
1
+
5
x
)
∼
5
x
(
1
+
u
)
r
−
1
∼
r
u
(
r
≠
0
)
(1+u)^{r}-1 \sim r\,u \quad(r\neq0)
(
1
+
u
)
r
−
1
∼
r
u
(
r
=
0
)
1
+
x
−
1
∼
x
2
\sqrt{1+x}-1 \sim \tfrac{x}{2}
1
+
x
−
1
∼
2
x
Casos discretos: sucesión aₙ, con n→∞
sin
(
1
n
)
∼
1
n
\sin\!\left(\tfrac1n\right) \sim \tfrac1n
sin
(
n
1
)
∼
n
1
n
sin
(
1
n
)
→
1
n\sin\!\left(\tfrac1n\right) \to 1
n
sin
(
n
1
)
→
1
ln
(
1
+
1
n
)
∼
1
n
\ln\!\left(1+\tfrac1n\right) \sim \tfrac1n
ln
(
1
+
n
1
)
∼
n
1
n
ln
(
1
+
1
n
)
→
1
n\ln\!\left(1+\tfrac1n\right) \to 1
n
ln
(
1
+
n
1
)
→
1
e
1
/
n
−
1
∼
1
n
e^{1/n}-1 \sim \tfrac1n
e
1/
n
−
1
∼
n
1
n
(
e
1
/
n
−
1
)
→
1
n\left(e^{1/n}-1\right) \to 1
n
(
e
1/
n
−
1
)
→
1
a
k
n
k
+
⋯
+
a
0
∼
a
k
n
k
(
a
k
≠
0
)
a_k n^{k} + \dots + a_0 \sim a_k n^{k} \quad(a_k\neq0)
a
k
n
k
+
⋯
+
a
0
∼
a
k
n
k
(
a
k
=
0
)
3
n
2
+
n
2
n
2
−
5
→
3
2
\dfrac{3n^{2}+n}{2n^{2}-5} \to \tfrac32
2
n
2
−
5
3
n
2
+
n
→
2
3
n
!
∼
2
π
n
(
n
e
)
n
(Stirling)
n! \sim \sqrt{2\pi n}\left(\dfrac{n}{e}\right)^{n}\ \ \text{(Stirling)}
n
!
∼
2
π
n
(
e
n
)
n
(Stirling)
n
!
n
n
→
0
\dfrac{n!}{n^{n}} \to 0
n
n
n
!
→
0