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Introduction to Limits

Limit is a concept that distinguishes calculus from other area of mathematics such as algebra and trigonometry. The concept of limit can be understood by considering examples such as finding the area of a region, slope of the tangent to a curve, the velocity of car and the sum of an infinite series.

The idea of limits give us a method for describing how the outputs of a function behave as the inputs approach some specified value. Definition and calculating limits of function values are explained. The calculation rules are straightforward and most of the limits we need can be found by substitution, graphical investigation, numerical approximation, algebra or some combination of these.

Neighbourhoods
Let a ∈ R.
If δ > 0, then the open interval (a – δ, a + δ) is called the δ-neighbourhood of 'a'.
i.e., {x ∈ R : a – δ < x < a + δ}
The location of (a – δ, a + δ) on the number line is





The set obtained by deleting the point 'a' from this neighbourhood is called the deleted neighbourhood of 'a'.
i.e., the deleted δ-neighbourhood of 'a' is
(a – δ, a) ⋃ (a, a + δ) or (a – δ, a + δ)\{a}.

Note:
i) Any interval (c, d) is a neighbourhood of some a ∈ (c, d).
Consider a = and δ = > 0

Then (a – δ, a + δ) =
= (c, d)
Therefore (c, d) is the δ-neighbourhood of 'a'.

ii) The set {x ∈ R : 0 < | x – a | < δ} is the deleted δ-neighbourhood of 'a', because
0 < | x – a | < δ ⇔ | x – a | < δ and x ≠ a
⇔ – δ < x– a < δ and x ≠ a
⇔ a – δ < x < a + δ and x ≠ a
⇔ x ∈ (a – δ, a + δ) and x ≠ a
⇔ x ∈ (a – δ, a + δ)\{a}

Limit point:
Let A be a subset of R.
A real number 'a' is said to be a limit point of A, if every neighbourhood of 'a' contains at least one element of A other than 'a'.

Limit of a function

Definition:
Let A be a subset of R, 'a' be a limit point of A and f : A → R.
A real number 'I' is said to be a limit of f at 'a', if to each given ε > 0,
∃ δ > 0 ∋ x ∈ A, 0 < | x – a | < δ
⇒ | f(x) – l | < ε.
It is denoted by f(x) = l or f(x) → l as x → a.

If the function is defined on a deleted neighbourhood of a real number 'a',
then the limit of the function at 'a' can be defined as follows:
Let a, l ∈ R and f be a function defined on a "deleted neighbourhood A of a".
Then l is said to be a limit of f at 'a' if
to each given ε > 0, ∃ δ > 0 ∋ x ∈ A, 0 < | x – a | < δ ⇒ | f(x) – l | < ε.
It is denoted by f(x) = l   or   f(x) → l as x → a.

A function f(x) is said to tend to a limit 'L' when 'x' tends to 'a', if the difference between f(x) and 'L' can be made as small as we please by making 'x' sufficiently near 'a'. We write:
f(x) = L or f(x) = L

The above is read as:
The limit of f(x) as 'x' approaches 'a' equals 'L'.
The notation means that the values of the function f(x) approach 'L' as the values of 'x' approach 'a' (but do not equal 'a').

Ex: Suppose f : (0, ∞) → R is defined by f(x) = √x.
Then f(x) = 0
Let ε > 0 be given. Choose δ = ε2. Then δ > 0.
For all x with x ∈ (0, ∞), 0 < | x | < δ i.e., 0 < x < δ
we have | f(x) – 0 | = √x < √δ = ε
∴ √x = 0

Two simple observations:
(i) For a constant value 'c',   (c) = c
(ii) For an identity function 'x',   (x) = a

Properties of Limits

By applying following six basic properties about limits, we can calculate many unknown limits.

Six basic properties
If P, Q, a, k and n are real numbers and
f(x) = P and g(x) = Q, then:
i) Sum rule:
[f(x) + g(x)] = f(x) + g(x) = P + Q
Therefore, the limit of the sum of two functions is the sum of their limits.

Ex: Find [x3 + x2 + x]
Sol: [x3 + x2 + x] = [x3] + [x2] + [x] = a3 + a2 + a.
ii) Difference rule:
[f(x) – g(x)] = f(x) – g(x) = P – Q
Therefore, the limit of the difference of two functions is the difference of their limits.

Ex: Find [x3 – x2 – x]
Sol: [x3 – x2 – x] = [x3] – [x2] – [x] = a3 – a2 – a.
iii) Product rule:
[f(x) * g(x)] = f(x) * g(x) = P * Q
Therefore, the limit of a product of two functions is the product of their limits.

Ex: Find [x2 (x + 2)]
Sol:
iv) Constant multiple rule:
[k.f(x)] = k. f(x) = k.P
Therefore, the limit of a constant times a function is the constant times the limit of the function.

Ex: Find [2x2 + 3x]
Sol: [2x2 + 3x] = [2x2] + [3x] = 2.[x2] + 3.[x] = 2a2 + 3a.
v) Quotient rule:
.
Therefore, the limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not equal to zero

Ex: Find
Sol:
vi) Power rule:
[f(x)]n = [ f(x)]n = Pn
Therefore, the limit of a power of a function is that power of the limit of the function.

Ex: Find (x2 + x)2
Sol: (x2 + x)2 = { (x2 + x)}2 = { (x2) + (x)}2 = {a2 + a}2
One-sided Limits : Right Hand Limit

Let f be a function and let a be any real number.
Then the right hand limit f(x)
represents the limit of function f as 'x' approaches a from right.

Algorithm for finding Right Hand Limit (RHL)
i. Write .
ii. Substitute x = (a + h) and replace x → a+ by h → 0 to obtain .
iii. Simplify by using the formula for the given function.
iv. The value obtained in above step is the RHL of f(x) at x = a.

Ex: Given f(x) = . Find .
Sol: Substituting x = 2 into the function f(x) leads to a '0' in both numerator and denominator.
Therefore, in order to calculate substitute x = (2 + h) and replace x → 2+ by h → 0.

Existence of a limit:
A function f(x) has a limit 'L' as 'x' tends to 'a' if and only if the right hand limit and left hand limit at 'a' exist and are equal. This can be written as:
f(x) = L f(x) = L and f(x) = L