Continuity and Derivatives:
Definition
A function f (x ) is continuous at x = c if lim
x →cf (x ) = f (c). Note that this
means all of the following three conditions must be met: f (c) is defined,
lim
x →cf (x ) exists,
lim
x →cf (x ) = f (c).
c c c c c
1,3 fail 3 fail 2,3 fail 1,2,3 fail 1,2,3 fail
| {z }
f (x ) is discontinuous at x = c
Theorem (Composition rule)
If lim
x →cg (x ) = L and f is continuous at x = L, then
lim x →cf g (x ) = f lim x →cg (x ) . Theorem
Limits at Infinity:
For some functions f (x ), limits such as lim
x →∞f (x ) andx →−∞lim f (x ) make
sense. Consider the function graphed below.
x y
2
−1 x −→x −→x −→ f (x )→ y = f (x )
As x moves to the right (towards positive infinity) the corresponding f (x ) value approaches 2. Such a limit is called a limit at infinity.
Definition (Limits at Infinity)
The statement lim
x →∞f (x ) = L means that for any real ε > 0, there is
a number N > 0 for which x > N implies |f (x ) − L| < ε.
The statement lim
x →−∞f (x ) = L means that for any real ε > 0, there is
a number N < 0 for which x < N implies |f (x ) − L| < ε.
Example
Investigate lim
x →∞
sin(x )
Sequences:
Recall that a sequence is an infinitely long list of real numbers a1, a2, a3, a4, a5, . . .
The number a1 is called the first term, a2 is the second term, a3 is the
third term, and so on. For example, the sequence 2, 34, 49, 165, 256, 367 , . . .
has nth term an an= n+1n2 . The nth term is sometimes called the general
A sequence an can be regarded as a function f : N → R, where
f (n) = an .
For example, the sequence 1 −1
n is the function f (n) = 1 − 1
n and here
is the graph of it.
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 · · · 1 1 2
Definition
A sequence an converges to a number L ∈ R provided that for any
ε > 0 there is an N ∈ N for which n > N implies |an− L| < ε.
Ifan converges to L, we denote this state of affairs as lim
n→∞an= L.
Definition
We say a sequence an diverges to ∞ if lim
n→∞an= ∞. This means
that for any L > 0, there is a positive N for which n > N implies an> L.
We say a sequence an diverges to −∞ if lim
n→∞an= −∞. This
means that for any L < 0, there is a positive N for which n > N implies an< L.
Example
