Linear Algebra and Calculus Lecture 1
Eric Sandin Vidal
November 25, 2024
Functions
Definition 1. Let D and S be sets. A function f : D → S is a rule that assigns a unique
element f (x) in S to each element in D.
Usually also written as
f :D → S
x → f (x) .
We call D the domain of f (D(f )). If x ∈ D, then we call f (x) the image of x. The set of
all the images of all the x ∈ D is called the range of f (R(f )). In this calculus part, unless
specified, functions will be f : R → R.
Example 2. Some examples of real functions are
• f (x) = x2 , with f (2) = 4, f (−2) = 4, f (3) = 9. Its domain is R and the range is R≥0 .
• f (x) = |x|, with f (−3) = 3, f (2) = 2, f (0) = 0. Its domain is R and the range is R≥0 .
• f (x) = sin x, with f (0) = 0, f (π) = 0, f (3π/2) = −1. Its domain is R and the range is
[−1, 1].
• f (x) = x1 , with f (1) = 1, f (2) = 12 , f (−5) = − 51 . Its domain is R\{0} and the range is
R\{0}.
Do not confuse a function with its graph, G(f ) = {(x, f (x)) ∈ R2 | x ∈ D}, which is the
set of points of the domain with its images.
Definition 3. If f, g are functions and x is in both their domains, then
• (f + g)(x) = f (x) + g(x)
• (f − g)(x) = f (x) − g(x)
• (f g)(x) = f (x)g(x)
(x)
• ( fg )(x) = fg(x)
, where g(x) ̸= 0.
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Definition 4. If f : D → S and g : R → D are functions, then f ◦ g : R → S is the
composition such that (f ◦ g)(x) = f (g(x)). The domain of f ◦ g is made of the values in the
domain of g whose image is in the domain of f .
Example 5. Let f (x) = cos x and g(x) = x3 . Then (f ◦ g)(x) = cos(x3 ).
Definition 6. A polynomial is a function P : R → R such that P (x) = a0 + a1 x + · · · + an xn ,
where a0 , a1 , · · · , an ∈ R are the coefficients with an ̸= 0 if n > 0. The number n is called the
degree of the polynomial.
Example 7. Some examples are
• P (x) = 3 is a polynomial of degree 0.
• P (x) = 2 − x is a polynomial of degree 1.
• P (x) = 3x3 + 2x2 − 1 is a polynomial of degree 3.
Proposition 8. Let P be a polynomial of degree n and Q a polynomial of degree m with n ̸= m.
Then:
• P + Q has degree max{n, m}.
• P Q has degree n + m.
P (x)
is a rational
Definition 9. Let P and Q be polynomials with Q ̸= 0, then R(x) = Q(x)
function whose domain is R except the values x such that Q(x) = 0.
2
+4
Example 10. The domain of R(x) = 3x
is R\{ 12 }.
2x−1
Definition 11. A zero of a function f is a value c ∈ R such that f (c) = 0.
A zero of a polynomial is also called root.
Example 12. A zero of f (x) = x3 − 3x2 + 2 is x = 1 since f (1) = 13 − 3 · 12 + 2 = 0.
Limits to a point
In this lecture we give non-formal definitions of limits.
Definition 13. We say that a function f has limit L at a point x = a if f (x) gets as close as
we want to L when we take x close enough to a and write it as
lim f (x) = L.
x→a
Example 14. Limits to some functions:
• limx→a x2 = a2
• limx→a c = c
Exercise 15. Find the following limits:
2
2
+x−2
• limx→−2 xx2 +5x+6
1
−1
• limx→a xx−aa
√
• limx→4 x2x−2
−16
Definition 16. If f is defined on (b, a) (resp. (a, b)), we define the left (resp. right) limit of f
as limx→a− f (x) (resp. limx→a+ f (x)) if x gets close to a from the left side (resp. right).
Example 17. Consider the function


1,
sgnx = 0,


−1,
x>0
x=0
x<0
Then limx→0− sgnx = −1 and limx→0+ sgnx = 1.
Theorem 18.
lim f (x) = L ⇐⇒ lim− f (x) = lim+ f (x) = L.
x→a
x→a
x→a
. What about limx→2 f (x)?
Exercise 19. Compute the lateral limits at x = 2 for f (x) = x2|x−2|
+x−6
Theorem 20. Let f, g be functions such that limx→a f (x) = L and limx→a g(x) = M and
k ∈ R, then
• limx→a (f (x) + g(x)) = L + M
• limx→a (f (x) − g(x)) = L − M
• limx→a (f (x) · g(x)) = L · M
• limx→a kf (x) = kL
(x)
L
• limx→a fg(x)
=M
, if M ̸= 0
m
m
• limx→a (f (x)) n = L n , provided L > 0 if n is even and L ̸= 0 if m < 0
• If f (x) ≤ g(x), then L ≤ M .
Theorem 21. (The Squeeze Theorem) Let f, g, h functions such that f (x) ≤ g(x) ≤ h(x)
∀x ∈ (b, c) and let a ∈ (b, c). Assume limx→a f (x) = limx→a h(x) = L. Then limx→a g(x) = L.
Example 22. Since 3 − x2 ≤ u(x) ≤ 3 + x2 ∀x ̸= 0, limx→0 3 − x2 = limx→0 3 + x2 = 3 =⇒
limx→0 u(x) = 3.
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Limits to ∞
Definition 23. Let f be defined over (a, ∞), if f (x) is as close to L by taking x large enough,
then we say limx→∞ f (x) = L. Analogously, limx→−∞ f (x) = M .
Exercise 24. Calculate limx→±∞ f (x), where f (x) = √xx2 +1 .
Example 25. For limits toward ∞ in rational functions it is important to look for the term
of highest degree:
2
• limx→±∞ 2x3x−x+3
= limx→±∞
2 +5
5
2x2 −x+3
x2
3x2 +5
x2
= limx→±∞
2− x1 + 32
x
3+ 52
x
= 32
+ 2
5x+2
x2
x3
• limx→±∞ 2x
= 0.
3 −1 = limx→±∞ 2− 1
x3
Example 26. Limits to points can also lead to infinity:
1
=∞
x→0 x2
lim
Example 27. There are also limits to infinity that lead to infinity
x2 + x3
x4 + 3x
=
lim
=∞
x→±∞ 1 − 1 + 52
x→±∞ x2 − x + 5
x
x
lim
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