CALCULUS
1. Limits and Continuity
Sequences
We will introduce the concept of limits for functions by a very useful
notion of a sequence. A sequence is a non-terminating list of real numbers
where each member of the sequence can be labeled by a natural number and it
is determined by some formula which depends on that label. For example, the
list of all positive even numbers 2, 4, 6, 8, … is a sequence since the n-th
member of the sequence (n is a natural number used to label the n-th element)
can be easily computed by the formula 2∙n. To make a notation concise, we may
often write {2𝑛}∞
𝑛=1 to denote this sequence.
1
Now consider a sequence which is given by the formula , where
𝑛
n can be any natural number. The first members of the sequence are
1 1 1
1, , , , … .Unlike the sequence above, this one has peculiar behavior. Higher
2 3 4
the number n is, closer the members of the sequence are to zero. We say that
1 ∞
zero is the limit of the sequence { }
1 ∞
We write { }
𝑛 𝑛=1
𝑛 𝑛=1
or that the sequence approaches zero.
→0
So far we have seen some sequences with limits and some sequences with
no limits. But can a sequence have more than one limit?
The answer is negative, every sequence has at most one limit. This can be easily
seen from the definition above. For example, the sequence
1,-1, 1,-1, …does not have two limits -1,1 the sequence simply does not have a
limit.
1 ∞
For example, we already know that { }
𝑛 𝑛=1
Since 0 <
1
1
𝑛
𝑛
→0 and {0}∞
𝑛=1 →0
1 ∞
for all n we have also that { 2 }
2 <
𝑛
𝑛=1
→0
Limits and functions
Now consider a set D which is a subset of the set of all real numbers. We say
that a real number d is a limit point of D if there is a sequence of elements from
D/{𝑑 } which approaches d. For examples, the half-open interval [0, 1) is a subset
of the set of all real numbers. Both 0 and 1 are limit points of [0, 1) and in fact
every number in [0, 1] is its limit point. On the other hand, the number 2 is not a
limit point of [0, 1) U {2}. Although a constant sequence {2}∞
𝑛=1 approaches
2,such a sequence is not allowed as 2 is excluded, note the formulation D \{𝑑} in
the definition of a limit point of a set. A point in the set D which is not its limit
point is called isolated.
Now consider a function f defined over a domain D. Let c be a limit point
of D. Consider a sequence 𝑐1 , 𝑐2 , …∈ D,𝑐𝑛 ≠ 𝑐 for all natural numbers n and
{𝑐𝑛 }∞
𝑛=1 → 𝑐. Now notice that 𝑓 (𝑐1 ) , 𝑓 (𝑐2 ), … is a sequence and we may ask
about its limit.
which gives the sequence
1. 92 , 1.992 , 1.9992 → 4
Also notice that𝑓(2) = 22 → 4. This indicates that the function does
not break at 2.
𝑥2
An example of a function which breaks at 0 is the rational function 𝑟(𝑥 ) =
𝑥
with the domain D = R \{0}. (This is simply because 𝑟(0) is not defined).
However, 0 is a limit point of D and that means we can create a sequence which
1 ∞
approaches 0 using the elements of D. For example { }
𝑛 𝑛=1
1
Now, 𝑟 ( ) =
𝑛
1
𝑛2
1
𝑛
defines a sequence. Since for every natural number n this expression is equal to
1
𝑛
1 ∞
and { }
𝑛 𝑛=1
1
∞
𝑛
𝑛=1
→ 0 we can conclude that limit of the sequence {𝑟 ( )}
𝑥2
is 0
The function 𝑟(𝑥 ) =
𝑥
with the domain D = R \ {0} is not continuous at 0 since r(0) is not defined (that
is 0 is not in the domain).However if we redefine function r to be the following
composite function:
𝑥 2 𝑖𝑓 𝑥 ≠ 0
𝑟(𝑥 ) { 𝑥
𝑖𝑓 𝑥 = 0
0
then this function is continuous. In order to make sure that we understand the
definition of a limit,we will play a bit with the following composite function.
No, it is not. The sequence -1,
1
1
2
3
−1 −1
2
,
3
, … approaches 0 too, however
𝑓(−1), 𝑓 (− ) , 𝑓 (− ) , … → −1 Moreover, the sequence
1
1
1
1
1
1
-1, , − , … → 0 but 𝑓(−1), 𝑓 ( ) , 𝑓 (− ) , 𝑓 ( ) … corresponds to
2
3 4
2
3
4
-1, 1, -1, 1; : : : which does not have any limit! Note the formulation every
sequence in the definition.
Notice that the function f breaks (is not continuous) at 0. This
creates a slightly different situation to the one where the rational function
𝑟(𝑥 ) =
𝑥2
𝑥
was considered. It also breaks at 0 but, on contrary,lim 𝑟(𝑥 ) exists
𝑥→0
and is equal to 0. In general, if a function r is not continuous at c but the limit at
c exists then the discontinuity could be ‘fixed’ simply by (re)defining 𝑟(𝑐) ≔
lim 𝑟(𝑥 ).
𝑥→𝑐
Rules for computing limits
Here we will introduce an effective methodology to compute limits.
Consider
lim
𝑥 3 −8
𝑥→2 𝑥−2
= lim
𝑥→2
(𝑥 2 +2𝑥+4)(𝑥−2)
𝑥−2
= lim (𝑥 2 + 2𝑥 + 4)
𝑥→2
The last limit above has in the argument a polynomial function so we can find
the limit simply by substituting 2 in place of x so as to get 22 + 2 ∙ 2 + 4 = 12.
Recall that polynomial functions are continuous. We can generalize the process
above in the following two rules.
2. Differentiation
Continuity and differentiability
Continuous at a Point
The function f is continuous at the point a in its domain if:
1. lim f(x) exists;
x→a
2. lim f(x) = f(a)
x→a
If f is not continuous at a, we say that f is discontinuous at a.
Note: If the point is not in the domain of f, we do not talk about whether or not
f is continuous at a.
Continuous on a Subset of the Domain
The function f is continuous on the subset S of its domain if it continuous at
each point of S.
Examples
1. All closed form functions are continuous on their (whole) domain. A closedform function is any function that can be obtained by combining constants,
powers of x, exponential functions, radicals, logarithms, and trigonometric
functions (and some other functions we shall not encounter here) into
a single mathematical formula by means of the usual arithmetic operations and
composition of functions. Examples of closed form functions are:
2x
3x2 - x + 1,
, and ex2-1.
x+3
2. The function f(x) = 1/x, also of closed form, is continuous at every point of its
domain. (Note that 0 is not a point of the domain of f, so we don't discuss what
it might mean to be continuous or discontinuous there.)
3. The function
f(x) =
-1 if x ≤ 2
x2+x if x > 2
is not a closed-form function (since we need two algebraic formulas to specify
it). Moreover, it is not continuous at x = 2, since limx→2f(x) does not exist.
3. Basic differentiation
Differentiation is an aspect of calculus that enables us to determine how one
quantity changes with regard to another.
Differentiation of a simple power
Differentiation of a unit power
Differentiation of a constant
Differentiation of a simple power multiplied by a constant
Differentiation of a unit power multiplied by a constant
Differentiation of a general power
Differentiation of a general power multiplied by a constant
Differentiation of a sum or difference of terms
Differentiation of a simple fraction
Differentiation of fractions reducible to simpler fractions
4. Advanced differentiation
Derivative Differentiability
The derivative of the function f at the point a in its domain is given by
lim
f'(a) =
h→0
f(a+h) - f(a)
h
We say that function f is differentiable at the point a in its domain if f'(a) exists.
Differentiable on a Subset of the Domain
The function f is differentiable on the subset S of its domain if it differentiable at
each point of S.
Note:
A function can fail to be differentiable at a
point a if either
lim
h→0
f(a+h) f(a) does not exist, or is
infinite.
h
In the former case, we sometimes have a cusp on the graph, and in the latter
case, we get a point of vertical tangency.
6. Lagrange mean value theorem
Lagrange's mean value theorem (often called "the mean value theorem," and
abbreviated MVT or LMVT) is considered one of the most important results in
real analysis. An elegant proof of the Fundamental Theorem of Calculus can be
given using LMVT.
Statement
Let
interval
be a continuous function, differentiable on the open
. Then
there exists some
such that
Informally, this says that a differentiable function must at some point grow with
instantaneous velocity equal to its average velocity over an interval.
Proof
We reduce the problem to Rolle's theorem by using an auxiliary function.
Consider
Note that
Rolle's theorem, there exists in
such that
which simplifies to
By
or $
as desired.