LIMITS
Introduction to The Notion of Limit
Limit basically describe how a function behaves near a specific point. The notion of limit of a
function is therefore about the behavior of the value of the function in the neighborhood of a
given point a. This notion is especially helpful to study the tendency of a function as the
variable x approaches infinity (∞) or negative infinity (-∞), using asymptotes for interpretation.
Furthermore, limits are fundamental for understanding continuity, derivatives, and concepts
related to curve length and area, which are further explored in Integration.
Definition:
If a function f is defined close to a real number a, and there is a value L such that f(x) comes
really close to it when x comes close enough to a, we say that f(x) has the limit L when x
approaches a. The function f has at most one left-hand limit and at most one right-hand limit
at a. In simple terms, this means that as x gets arbitrarily close to a from the right or left, f(x)
gets arbitrarily close to L.
lim 𝑓(𝑥) = 𝐿
𝑥→𝑎
A formal definition statement for limit can be made by the epsilon-delta definition, where
epsilon stands for the variable ε, and delta for δ. As above lim (x→ a) f(x) = L if and only if,
for each ε > 0, there exists a δ > 0 such that, for all x ≠ a between a − δ . and a + δ, the value
f(x) lies between L − ε and L + ε.
We say that the function f(x) approaches L arbitrarily close from the left or right of a if, for each
small positive value ε some positive δ, such that the absolute difference between f(x) and L
(|f(x) - L|) is less than ε for the open interval (a, a - δ) or (a, a + δ).The absolute value |f(x) - L|
< ε expresses that the function's value is within a small range (ε) around the limit L.
Start with a desired upper bound ε on the distance from f(x) to L. The value f(x) must then lie
in the horizontal strip bound by the lines y = x − L and y = x + L. The statement on the limit
now says that we can indicate an upper bound δ on the distance from x to a so that this
happens. In other words: if x is chosen between a − δ and a + δ, then the value of f(x) will end
up within the desired distance ε to L.
With the help of quantifiers and intervals the formal definition statement provided by the
epsilon-delta definition can also be written as above. In here:
• ∀ε > 0 stands for "for all positive numbers"
(think about an arbitrarily small positive number)
• ∃δ > 0 stands for "there is a positive number δ"
(think about a positive number that is chosen sufficiently small)
• (a − δ, a + δ) ∖ {a} stands for the set obtained by removing the point a from the open
interval (a − δ, a + δ)
(hence, these are the points at distance at most δ to a distinct from a)
• p ⇒ q stands for "if p is true, then also q"
• f(x) ∈ (L − ε, L + ε) stands for "f(x) lies at distance less than ε to L"
(this can also be written as |f(x) − L| < ε ).
The limit only exists if the left-hand and right-hand limit are equal at a. We define continuity
by use of the notion of limit such that a function f(x) is continuous at the point a if the limit
exists at a and the limit is equal to the function of a at that point. A function is generally
continuous if continuous on the entire domain of the function.
lim 𝑓(𝑥) = 𝑓(𝑎)
𝑥→𝑎
Limits and Infinity:
So far we looked at the limit L of a function at a point a where L and a are real numbers (R).
However, sometimes f(x) becomes arbitrarily small or large or we want to know what happens
to f(x) as x gets arbitrarily small or large. For the limits of these functions one or both of L and
a can be infinite as in:
lim 𝑓(𝑥) = ∞
𝑥→𝑎
lim 𝑓(𝑥) = −∞
𝑥→𝑎
lim 𝑓(𝑥) = 𝐿
𝑥→∞
lim 𝑓(𝑥) = 𝐿
𝑥→−∞
The symbols ∞ and −∞ , or infinity and minus infinity, respectively are concepts representing
virtual numbers greater or smaller than all the real numbers (R). They are not real numbers,
but to some extent we can calculate with these symbols. Working with infinity as an arbitrarily
large number can therefore be useful, but has its restrictions. Dividing by zero is undefined, as
we know. In the same vein, there are some mathematical expressions with infinity that remain
undefined. Thus:
1
=0
∞
1
∴ lim = 0
𝑥→∞ 𝑥
If we multiply both sides of 1 / ∞ = 0 by ∞, the two ∞ symbols cancel each other out, and we
find 1 = 0 ⋅ ∞. This equation does not make sense. For, if we do the same calculation starting
from 2 / ∞ = 0, we get 2 = 0 ⋅ ∞, which leads to the contradiction 1 = 2. Therefore, we can make
sense of 1 / ∞ but not of 0 ⋅ ∞.
If lim (x→a) f(x) = ∞, then the limit is not real in the earlier definition of limit because infinity is
not a number. We distinguish three cases for limits:
1. the limit is a real number
2. the limit equals infinity or minus infinity
3. the limit does not exist (that is, is neither a real number nor ±∞)
A more precise definition of lim (x→a) f(x) = ∞ is: for each positive number n there is a positive
number δ such that for all numbers x distinct from a at distance less than δ to a, the value of
f(x) is greater than n.
Again, with the help of quantifiers and intervals this statement can be written as above. Here:
• ∀n ∈ N denotes "all natural numbers"
(think of an arbitrarily large number)
• ∃δ > 0 denotes "there is a positive number δ"
(think of a positive number which is chosen to be sufficiently small)
• (a − δ, a + δ) ∖ {a} denotes the set that arises from removing the point a from the
open interval (a − δ, a + δ)
(these are the points at distance strictly smaller than δ to a and distinct from a).
We use the limit to infinity when we want to know what happens to f(x) as x becomes
arbitrarily small or large. In such cases we do not need to mention the left-hand or right-hand
limit as there are no numbers to the right of ∞, nor to the left of −∞. The same discontinuous
function as above, can be examined using limit to infinity. It then becomes an expression of a
common limit as shown below left. The common limit, lim (x→±∞) c / x^n = 0, will often be
used, where c and n are real numbers (R) and n > 0. The value of this limit is 0, because we
divide a constant by a number that increases in absolute value, which causes the value of the
fraction to approach 0.
𝑐
=0
𝑥𝑛
𝑥 → ±∞
lim
The case of twice infinity comes into play when we consider L = ∞. Then we are dealing with
the equation lim (x→∞) f(x) = ∞ in which ∞ occurs twice. We then say that the function f(x)
has limit ∞ for x approaching ∞. In this case, the function value f(x) gets arbitrarily larger as x
gets larger and larger. There are two other variations of the equation lim (x→ ∞) f(x) = ∞, in
which one of the two occurrences of ∞ has been replaced by −∞.
Calculation Rules for infinity (±∞), assume the usual laws like associativity, distributivity, and
commutativity. The comparison of limits with infinity and those without becomes easier if we
work with intervals like (n, ∞). This example stands for the set of all real numbers x with x > n.
The statement that f(x) tends to ∞ for x → a can then be phrased as: for each real number n,
there is a positive number δ such that, for all x ∈ (a − δ, a + δ) ∖ {a}, we have f (x) ∈ (n, ∞).
Some Examples:
• −5 + ∞ = ∞
• 6 − ∞ = −∞
• −7 ⋅ ∞ = −∞
•
•
•
•
−∞ ⋅ ∞ = −∞
4/∞=0
∞ / 100 = ∞
∞ ^−3 is undefined
•
With Rule 7 the reason for ∞^a = 0 when a < 0 is that, when infinity (∞) is raised to a
negative power, the result is 0 in the following manner: Firstly, dealing with a negative
power is equivalent to taking the reciprocal and then raising to the absolute value of
the power. Secondly, the reciprocal of infinity (∞) is 0 and 0 raised to any positive power
is 0.
•
Regarding undefined Iimits; in order to see whether a rule is defined, and if so, why it
is true, it is useful to replace ∞ by the limit of a function. The rules for calculating limits,
discussed later, will then offer a solution.