9-2 Esercizi sui limiti

9
Esercizi su limiti
1. Verificare mediante la definizione i seguenti limiti.
(a)
(b)
lim 5x−12=3
x 3
lim
x ∞
3x−2
3
=
2x1
2
(d)
lim
(e)
lim
x ∞
x ∞
x
= 0
x 1
2
x2
= ∞
x−1
2
(c)
lim
x ∞
x −4
= 1
x −5x6
2
2. Applicando i teoremi sull'algebra dei limiti calcola i seguenti limiti:
(a)
lim  x 2 x 
1
x 1
(b)
lim
(c)
lim x⋅ln x 
x ∞
2
x ∞
3. Studiare in funzione del parametro
(a)
lim p x
(b)
lim
(c)
p
lim
x ∞
1− p x
1− p
x 2p1
p
x 1
4. Calcolare i seguenti limiti:
(a)
lim
x ∞
x
sen x 
(e)
lim
cos x 
x
(f)
lim
x
sen  x 
i seguenti limiti:
x ∞
x ∞
lim
(d)
x 3
3 x3 x1
2
2
2 x −3 x x
1/3

x
2
x ∞
x ∞
(b)
lim
5 x 4 −1
5
3
3 x −2 x x
(c)
lim
7 x 3x 2 1
3
2
6 x −8 x 14
(d)
lim   9 x 6 x−1−3 x
1
(e)
lim  9 x −x 1−2 x 
∞
(f)
lim
2x 3− x 2 −5x−2
2
2x −5x2
5 
(g)
lim
3x 2 x−10
2
x −5x−14
(h)
lim
(i)
lim
tg x
x
1
(j)
lim
e 2x −1
x
2
lim
ex −e−x
8x

x ∞
x 0
2
x ∞
2
x ∞
x 2
x 2
x 0
x 0
x 0
(k)
 3 x 3−x 4−2
 2 x25x9−3
x 0
 
−
1
4
2
(l)
e 2 x −e 2
lim
2
x 0 1−cos x
(m)
lim
x ∞

3x−1
3x2

x
2
e2
1
e
2/3
3
10
Appendice
Limiti notevoli e limiti particolari
a)
lim
sen x
=1
x
b)
lim
1−cos x 1
=
2
2
x
c)
lim 1
d)
lim
ex −1
=1
x
e)
lim
log1 x
=1
x
f)
lim 1
x 0
x 0
x
x ∞
x 0
x 0
 
1
=e
x
x
 
1
lim  1  x=e
x
x ∞

=e 
x

g)
h)

x ∞
1
x
lim  1x  =e
x 0
i)
lim
x 0
j)
lim
k)
lim
x ∞
x 0
log a 1 x
1
=
x
ln a
ln  x 
=0
x
e  x−1
=ln 
x
3/3