Exercise Lecture of Mathematics
November 15, 2016
Exercise 1. Solve the following inequalities:
2
a) log 1 (4 − 9x ) ≥ −1
b) log |x − 1| < log x + log 2
c) log2 x + log8 x < 1
d) log3 |3 − 2x| ≥ − log 1 (1 + x )
3
e) 2
|x+2|
≤ 32 · 2
2
3
x2 −1
2x
f ) 3 |x+2| < 9
Solutions
�
� �
a) − 23 , − 13 ∪ 13 ,
√
4
c) ]0, 8[
�
�
e) − 73 , +∞
2
3
�
�
�
∪ 1, +∞
√
√
d) [−(1 + 3), 3 − 1]
√
√
f ) ]1 − 6, 1 + 6[
b)
�1
3,1
�
Exercise 2. Determine the domain of the following functions:
1
log x
�
�
x
d) f (x) = log 1 √
x−2
2
2
a) f (x) =
b) f (x) = log4 (x + 1)
e) f (x) =
e−x
cos(2x)
f ) f (x) = log
h) f (x) =
1
2x−4 − 7
i) f (x) =
x(x+2)
g) f (x) = e (x+1)3
c) f (x) = log
�
�
x2 + 4
x
�
2
arcsin(x − 1)
4 + cos x
(3 − e2x )4
Solutions
b) R
a) ]0, 1[ ∪ ]1, +∞[
c) ]0, +∞[
�
e) R \ π
4 +
kπ
2
:k∈Z
d) ]4, +∞[
√
√
f ) [− 2, −1[ ∪ ]1, 2]
�
g) ] − ∞, −1[ ∪ ] − 1, +∞[
√
√
i) ] − ∞, log 3[ ∪ ] log 3, +∞[
h) ] − ∞, 4 + log2 7[ ∪ ]4 + log2 7, +∞[
Exercise 3. Consider the following functions:
�
�
g: − π
2 , +∞


tan x
x2
g(x) =

 4
log2 x
f :R → R,
f (x) =
�
−(x2 + 1)
e3x
if x < 0,
if x ≥ 0.
For each of them, answer to the following questions.
a) Compute the limits at the extreme points of the domain.
b) Is the function continuous in the whole domain?
c) Is the function monotone?
1
→ R,
if −
π
2
< x ≤ 0,
if 0 < x ≤ 2,
if 2 < x.
�
d) Determine the image of the function.
e) If possible, compute the inverse function.
Finally draw the graph of f and g.
Solutions
Study of the function f
a)
lim
x→−∞
f (x) = −∞,
lim
x→+∞
f (x) = +∞.
b) The function f is continuous in R \ {0}: 0 is a point of discontinuity since
−1 �= 1 = lim f (x).
lim f (x) =
x→0−
x→0+
c) The function is monotone increasing.
d) − e) We notice that
f (R) = ] − ∞, −1[ ∪ [1, +∞[.
Then, naming f¯ : R → ] − ∞, −1[ ∪ [1, +∞[ given by f¯ = f , we have that f¯ is injective,
since it is monotone, and surjective, then bijective, that is, invertible, and the inverse
function is f¯−1 : ] − ∞, −1[ ∪ [1, +∞[ → R,
−1
f¯ (y) =
� �
− −(1 + y)
1
3 log y
The graph of f is the following.
2
if y < −1,
if y ≥ 1.
Study of the function g
a)
lim
x→− π +
2
g(x) = −∞,
lim
x→+∞
g(x) = +∞.
b) The function g is continuous in its domain.
c) The function is monotone increasing.
d) − e) We notice that
g
��
−
π
2 , +∞
��
= R,
then the function is surjective; since it is also injective, �by monotonicity,
g is bijective,
�
that is, invertible, and the inverse function is g −1 : R → − π
2 , +∞ ,
g
−1


y
arctan
√
(y) =
4y

 2y
The graph of g is the following.
3
if y < 0,
if 0 ≤ y < 1,
if 1 ≤ y.
Exercise 4. The following limits are indeterminate forms. Compute them by applying suitable
relevant limits.
a)
d)
lim
x→−∞
x5 + 2x3 + 1
x2 + 7x + 4
b)
−x
e)
3
lim (x + 3x)2
x→+∞
x2 + sin x
x2 + 2x − 5
lim
x→−∞
2
x
lim (x + 3 − log3 x)3
x→+∞
sin 5x
g) lim
x→0
7x
sin(x2 + x)
h) lim
x→0
x
j) lim sin x log6 x
k) limπ
x→0+
m) lim
x→0
x sin x
cos x − 1
2
esin x − 1
p) lim
√
x→0 1 −
x2 + 1
x→
4
1 − tan x
sin x − cos x
n) lim √
x→0
ex − 1
1 − cos x
log(cos x)
q) lim
x→0
x2
x2 + 1
5x
�
� 1
f ) lim 1 + x 2x
c)
−x
lim
x→+∞
x→0
√
1 − cos x
x
x→0+
√
sin( x − 1)
l) lim
x→1
x−1
i) lim
o) lim
x→0
ex sin x − 1
1 − cos x
r) lim
1
1+x
Solutions
a) − ∞
b) 1
d) 0
e) 1
5
g)
7
h)1
j) 0
k) −
m) − 2
p) − 2
c) 0
√
f)
√
2
n) it does not exist
q) −
1
2
e
1
i)
2
1
l)
2
o) 2
r) − 1
4
− cos x
x
x→0