limites notaveis

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Calculo de limite Prof. Eva Figueiredo 
Limites Notáveis!!! 
 
Indeterminações 
∞ − ∞ , 
∞
∞
, 
0
0
, 0× ∞ 
Recorda, para 1a > , 
 
( )lim 0x
x
a
→−∞
= ( )lim x
x
a
→+∞
= +∞ ( )
0
lim loga
x
x
+→
= −∞ ( )lim loga
x
x
→+∞
= +∞ 
lim ,
x
px
a p
x→+∞
= +∞ ∈ℝ lim 0 ,
p
xx
x p
a→+∞
= ∈ℝ 
( )loglim 0 , 0apx
x
p
x→+∞
= > ( )lim , 0log
p
x
a
x p
x→+∞
= +∞ >
 
0
1lim 1, 0
bx
x
e b
bx→
−
= ≠ 
( )
0
ln 1
lim 1 , 0
x
bx
b
bx→
+
= ≠ 
 
 
Limites Notáveis: 
lim ,
x
px
e p
x→+∞
= +∞ ∈ℝ 
( )lnlim 0
x
x
x→+∞
= 
0
1lim 1
x
x
e
x→
−
= 
( )
0
ln 1
lim 1
x
x
x→
+
= 
 
 
Usando os limites indicados, calcula os seguintes limites: 
 
1. 
2
0
lim
5
x x
x
e e
x→
−
 2. 
3 3
0
lim
h
h
e e
h
+
→
−
 3. ( )
1
lim 2 1 1n
n
n e
→+∞
 
+ − 
 
 
4. 
2
0
2lim
1
x
x
x
x→
+
+
 
5. 1lim
xx xe→−∞
 6. 
1
2
0
lim x
x
x e
+→
 
7. ( )
0
ln 2 ln 2
lim
2x
x
x→
+ −
 8. ( )
3
0
ln 1
lim
2x
x
x→
+
 
9. ( )
2
3
5 6lim
ln 2x
x x
x→
− +
−
 
10. ( )
0
log 2 log 2
lim
2x
x
x→
+ −
 11. ( )
0
ln 1 3
lim
4x
x
x→
+
 12. ( )
5
ln 6
lim
5x
x
x→−
+
+
 
13. ( )ln 3lim
x
x
x→+∞
 14. ( )
2
0
1lim
ln 3 1
x
x
e
x→
−
+
 
15. 1lim 2 .ln 1
x
x
x→+∞
 
+ 
 
 
17. ( )ln 3lim
1x
x
x→+∞
+
+
 
18. ( )ln 1
x
lim x x
→+∞
+ −   19. ( )1xlim f x→ , sendo 
 ( ) ( )
22
 , 1
4 1
ln 2 1
, 1
1
x x
x
xf x
x
x
x
 − ≤ −
= 
− >

−