To each ϵ > 0, there exists a δ > 0 such that
|f(x) − a| < ϵ whenever 0 < |x − x0| < δ.
The advantage of this definition is that it does not require any
intuition of how big or how small ϵ or δ should be. Example:
f(x) =
⎧ ⎪ ⎨
⎪ ⎩
x2
x ≠ 2
8
x = 2
We claim that limx→ 2f(x) = 4, not 8 as for
an arbitrarily given ϵ > 0, one can always choose δ = √{4 + ϵ} − 2 even though f(2) = 8.
In engineer's language, it suffices to say that limx→x0f(x) = a means that f(x) can be arbitrarily close to a when
x is sufficiently close to x0 (Note: x does not have to be
equal to x0).
If limx→ x0f(x) exists and is equal to f(x0), we
say that f(x) is continuous at x = x0. Obviously, if f(x) is
continuous at x0, it has a limit as x→ x0 but the
converse is not true. Example
Show that
lim x→ 0
f(x) = 0,
if
f(x) ≡
1
x
−
1
sin x
.
Theorem (L'Hospital):
If f(a) = g(a) = 0, f(x) and g(x) are differentiable at x=a, and g′(x) ≠ 0, then,
lim x→ a
f(x)
g(x)
=
lim x→ a
f′(x)
g′(x)
(2)
(Proof):
lim x→ a
f(x)
g(x)
=
lim x→ a
f(x) − f(a)
g(x) − g(a)
=
lim x→ a
f(x) − f(a)
x − a
g(x) − g(a)
x − a
=
f′(a)
g′(a)
.
(3)
Note that the above theorem is also applicable when you have
[(∞)/(∞)] (choose f*(x) = 1/f(x)). Some exercise
(a)
lim x→ 0
1 − cos x
x2
(b)
lim x→ ∞
ex
xn
(c)
lim x→ +0
xx
(d)
lim x→ 0
cos x
x + 1
Another example of somewhat exotic function (Dirichlet function):
f(x) =
lim m→ ∞,n→ ∞
cos 2n(m! πx).
(4)
As shown in class, this function is 1 when x is a rational number
and 0 when x is an irrational number. It is difficult
to sketch this function.
Any usage of such a function ? This is an example of functions that
is integrable in the sense of Lebesgue but not in the sense of Riemann.
What is
⌠ ⌡
1
0
f(x) dx
?
Riemann integrals and Lebesgue integrals.
Measure
Cantor's diagonal argument
ℵ0, ℵ1, … (aleph)
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version 4.03. On 15 Aug 2023, 23:06.