5Differentiability
IA Analysis I
5.1 Limits
First of all, we need the notion of limits. Recall that we’ve previously had limits
for sequences. Now, we will define limits for functions.
Definition (Limit of functions). Let A ⊆ R and let f : A → R. We say
lim
x→a
f(x) = `,
or “f(x) → ` as x → a”, if
(∀ε > 0)(∃δ > 0)(∀x ∈ A) 0 < |x − a| < δ ⇒ |f(x) − `| < ε.
We couldn’t care less what happens when
x
=
a
, hence the strict inequality
0 < |x − a|. In fact, f doesn’t even have to be defined at x = a.
Example. Let
f(x) =
(
x x 6= 2
3 x = 2
Then lim
x→2
= 2, even though f(2) = 3.
Example. Let f(x) =
sin x
x
. Then f (0) is not defined but lim
x→0
f(x) = 1.
We will see a proof later after we define what sin means.
We notice that the definition of the limit is suspiciously similar to that of
continuity. In fact, if we define
g(x) =
(
f(x) x 6= a
` x = a
Then f(x) → ` as x → a iff g is continuous at a.
Alternatively,
f
is continuous at
a
if
f
(
x
)
→ f
(
a
) as
x → a
. It follows also
that
f
(
x
)
→ `
as
x → a
iff
f
(
x
n
)
→ `
for every sequence (
x
n
) in
A
with
x
n
→ a
.
The previous limit theorems of sequences apply here as well
Proposition. If f(x) → ` and g(x) → m as x → a, then f(x) + g(x) → ` + m,
f(x)g(x) → `m, and
f(x)
g(x)
→
`
m
if g and m don’t vanish.