b) Utilizzo di alcuni limiti notevoli.
Esercizi : calcolare i seguenti limiti :
( )
=
− + +
=
− + +
=
+ + −
⇒
=
+
− +
− + −
=
+ + −
−
⇒
=
+
−
+
⋅ −
−
⋅ +
+∞
→ +∞
→ +∞
→
+∞
→ +∞
→
∞ +∞
→
3 4 4
3
4 3 1 1
lim 4
3 1 1
3 lim 1 4
lim
3 1 1 3
3 lim 1 1
1 lim 3 1
lim 1
x x x
x x
x x
x
x x x
x x x
x x x
x
x x
x
x x
x
x e
x
x
=
+
+∞
→
1 1
lim ( ) x x e
x
+ =
→
1 1
lim
0x
ae
x a
x
1 log 1 log
lim =
+
+∞
→ a
( ) x ae
x
x log
1 log
lim 1
0
+ =
→
x a a x
x
1 ln
lim
0− =
→
( 1 )
lim − 1 = + ∞ >
+∞
→
a
x a x
x
( 0 1 )
1 0
lim − = < <
+∞
→
a
x a x
x
( 1 )
1 0
lim − = >
−∞
→
a
x a x
x
( 0 1 )
lim − 1 = − ∞ < <
−∞
→
a
x a x
x
INDICE
4 3
4
4 3 3
4 4
3
4 1 3 1 lim 4
1 3 1
lim −
+
−
− + +
⋅ −
−
⋅ +
=
− + +
=
− + +
⇒
→+∞ →+∞e
x x x
x x x
x
x
xx
ricordando che : e
x
x
x
=
+
+∞
→
1 1
lim .
Allo stesso modo si poteva procedere utilizzando la trasformazione : f ( ) x g ( ) x = e g ( ) x ln f ( ) x
( )
( )
( )( )
3 4 2 4
1 3 1 4
1 3 3
1 3
1 3 ln 1
3 ln 1 3 1
ln
2 2
2 2
2
lim
lim lim lim
lim 3 1
lim 1
− − + +∞ −
→
−
− + +∞
− →
−
⋅ + +
+
− + +∞
→
+
− +∞
→
+
⋅ − +∞
+ − →
⋅
=
⇒
=
=
=
⇒
=
=
⇒
=
+
−
∞ +∞
→
e e
e e
e
e e
x x
x x
x x
x x x
x x x
x x
x H x
x x
x
x x x x
x x x x
x x
x
N.B. abbiamo anticipato l’applicazione di l’Hospital ( trattato però successivamente! )
3 2 2
2
3 3 3
3
3 1 1 lim 3
1 1 3 lim
1 lim
1 3 3 lim
1 3 lim
3 1 1 3 lim
1 lim
e
x x
x
x x
x
x x x
x x
x x
x
x x
x
x x
x x
=
+
=
+
=
+
⇒
=
+
⋅
+
=
+
⋅
+
=
+
+∞
→
⋅
+∞
→ +∞
→
+∞
→ +∞
→ +∞
→ +∞
→
+
INDICE
calcolare :
( ) 1
2 lim 1
2 lim
2 2 1 lim 2
2 1 1 lim
lim
2 3 3
2 3 3
3 2 2
0 0
0 0
0
=
−
⋅
⇒
=
⋅ −
=
−
=
−
→ +
→+
→+
→+
→ +
x x x
x x x
x x x
x a
x a x
a x
a
x x
x x
x
calcolare il valore del parametro reale a per cui sia :
( )
a a
a a ctgx
a ctgx
tgx tgx
tgx tgx
e a
ctgx a
ctgx
ctgx a ctgx
a ctgx
a ctgx
a ctgx
a
x x
x x
x x
2 2
2 2 2
1 2 1
2 1 2 1
2 1 1 lim 2
1 1 lim
1 2 2 lim
2 1 1 lim 2 1
1 lim
2 2 1 lim
0 0
0 0
0 0
=
+
=
+
⇒
+
=
+
⋅
+
⇒
=
+
=
+
→ +
→+
→ +
→ +
∞
→ +
→ +
⋅ +
+
da cui eguagliando : e 2 a = 2 ⇒ 2 a = ln 2 ⇒ a = ln 2
INDICE
