Limit of a Sequence. Limit of a Function

1.

CALCULUS · PRESENTATION
Lim it o f a S eq uence.
Lim it o f a Functio n
Definitions, main theorems, remarkable limits and the table of
derivatives
Presented by: [Name Surname], [group]
Teacher: [Teacher’s name]
Tashkent — 2026
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CONTENTS
Outline
01
What is a number sequence
02
Limit of a sequence
03
Limit of a function: Cauchy and Heine
04
Theorems on limits
05
Remarkable limits and examples
06
The derivative and its table
07
Conclusion
Limit of a Sequence. Limit of a Function
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01 · THE IDEA
W hat is a num b er seq uence?
If every natural number n = 1, 2, 3, … is matched by some rule with exactly one number aₙ, the list a₁, a₂,
a₃, …, aₙ, … is called a number sequence.
aₙ is the general term (the n-th term) of the sequence.
aₙ = 1/n
aₙ = n²
aₙ = (−1)ⁿ
1; 1/2; 1/3; 1/4; …
1; 4; 9; 16; …
−1; 1; −1; 1; …
The terms get closer to 0
The terms grow without bound
The terms jump between −1 and 1
Limit of a Sequence. Limit of a Function
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4.

02 · LIMIT OF A SEQUENCE
Definition of the limit of a sequence
Definition. A number a is called the limit of the sequence
(aₙ) if for every ε > 0 there is a natural number N such that
|aₙ − a| < ε for all n > N.
aₙ
a+ε
a=1
lim aₙ = a
a−ε
n→∞
n
N
aₙ = 1 + (−1)ⁿ/n, ε = 0.2 gives N = 5: all terms with n > 5 (orange)
are inside the ε-band.
Limit of a Sequence. Limit of a Function
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5.

02 · LIMIT OF A SEQUENCE
Convergent and divergent sequences
Convergent
Divergent
Has a finite limit
Has no finite limit
1/n → 0
n² → ∞
(n + 1)/n → 1
√n → ∞
1/2ⁿ → 0
(−1)ⁿ: −1; 1; −1; … — no limit
Uniqueness. A sequence cannot have Boundedness. Every convergent
two different limits.
sequence is bounded.
Limit of a Sequence. Limit of a Function
Weierstrass. A monotonic bounded
sequence converges.
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03 · LIMIT OF A FUNCTION
Limit of a function (Cauchy definition)
Definition. A number A is the limit of f(x) as x → x₀ if for
every ε > 0 there is δ > 0 such that |f(x) − A| < ε for all x
with 0 < |x − x₀| < δ.
y
x₀ − δ
x₀ + δ
f(x)
A+ε
A
lim f(x) = A
A−ε
x→x₀
x
x₀
When x is in the blue δ-band, f(x) stays in the orange ε-band.
f(x₀) itself does not have to exist.
Limit of a Sequence. Limit of a Function
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03 · LIMIT OF A FUNCTION
Heine definition and one-sided limits
Heine definition (via sequences)
One-sided limits
A is the limit of f(x) as x → x₀ if for every sequence
(xₙ) with xₙ → x₀ (xₙ ≠ x₀) the sequence of values
f(xₙ) → A.
Left limit:
The Cauchy and Heine definitions are equivalent. Heine’s
definition connects the limit of a function with the limit of a
sequence.
Limit of a Sequence. Limit of a Function
lim
f(x)
x→x₀−0
Right limit:
lim
f(x)
x→x₀+0
The limit exists ⇔ both one-sided limits exist and
are equal.
Example: f(x) = |x|/x. As x → 0 the left limit is −1 and the right limit is
1, so the limit does not exist.
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04 · THEOREMS
Main theorems on limits
If lim f(x) = A and lim g(x) = B (as x → x₀), then:
lim C = C
The limit of a constant is the constant itself
lim (C · f(x)) = C · A
A constant factor can be taken out of the limit
lim (f(x) ± g(x)) = A ± B
The limit of a sum (difference) is the sum (difference) of the limits
lim (f(x) · g(x)) = A · B
The limit of a product is the product of the limits
lim (f(x) / g(x)) = A / B, B ≠ 0
The limit of a quotient is the quotient of the limits (if the limit of the denominator
is not zero)
These theorems are also true for sequences (n → ∞).
Limit of a Sequence. Limit of a Function
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05 · REMARKABLE LIMITS
The two remarkable limits
FIRST REMARKABLE LIMIT
SECOND REMARKABLE LIMIT
sin x
lim
=1
x
lim (1 + 1/n)ⁿ = e
x→0
For small x, sin x ≈ x (x in radians).
n→∞
e ≈ 2.718 — Euler’s number, the base of the natural logarithm.
They are used to derive the table of derivatives: (sin x)′ = cos x and (eˣ)′ = eˣ.
Limit of a Sequence. Limit of a Function
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05 · EXAMPLES
Examples of calculating limits
Indeterminate forms: 0/0, ∞/∞, ∞ − ∞, 0 · ∞ — we resolve them by transforming the expression.
0/0 form
∞/∞ form
1st remarkable limit
lim (x² − 4)/(x − 2)
lim (3n² + 1)/(n² − 5)
lim sin 5x / x
x→2
n→∞
x→0
= lim (x − 2)(x + 2)/(x − 2)
= lim (3 + 1/n²)/(1 − 5/n²)
= lim 5 · sin 5x / 5x
x→2
= lim (x + 2) = 4
x→2
Factor the numerator and cancel (x − 2).
Limit of a Sequence. Limit of a Function
n→∞
x→0
= (3 + 0)/(1 − 0) = 3
=5·1=5
Divide the numerator and denominator by
n²: 1/n² → 0.
Let t = 5x: as x → 0, t → 0 and sin t / t → 1.
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06 · DERIVATIVE
From the limit to the derivative
The derivative is the limit of the ratio of the increment of the function to the increment of the argument:
f(x + Δx) − f(x)
Δy
f′(x) = lim
= lim
Δx
Δx→0 Δx
Δx→0
Example: f(x) = x²
Meaning
Δy = (x + Δx)² − x² = 2x · Δx + Δx²
Geometric: f′(x₀) is the slope of the tangent line to the graph at the
point x₀, f′(x₀) = tan α.
Δy / Δx = 2x + Δx → 2x as Δx → 0
(x²)′ = 2x
Limit of a Sequence. Limit of a Function
Physical: the derivative of distance with respect to time is velocity,
s′(t) = v(t).
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06 · DERIVATIVE
Table of derivatives
FUNCTION
DERIVATIVE
FUNCTION
DERIVATIVE
C (constant)
0
sin x
cos x
x
1
cos x
−sin x
xⁿ
n · xⁿ⁻¹
tan x
1 / cos² x
√x
1 / (2√x)
cot x
−1 / sin² x
1/x
−1 / x²
arcsin x
1 / √(1 − x²)
aˣ
aˣ · ln a
arccos x
−1 / √(1 − x²)
eˣ
eˣ
arctan x
1 / (1 + x²)
logₐ x
1 / (x · ln a)
arccot x
−1 / (1 + x²)
ln x
1/x
Limit of a Sequence. Limit of a Function
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06 · DERIVATIVE
Rules of differentiation
(u ± v)′ = u′ ± v′
Sum and difference rule
(C · u)′ = C · u′
A constant factor can be taken out of the derivative
(u · v)′ = u′v + uv′
Product rule
(u / v)′ = (u′v − uv′) / v²
Quotient rule (v ≠ 0)
(f(g(x)))′ = f′(g(x)) · g′(x)
Chain rule (composite function)
EXAMPLES
(x³ + sin x)′ = 3x² + cos x
Limit of a Sequence. Limit of a Function
(x² · eˣ)′ = 2x · eˣ + x² · eˣ
(sin 3x)′ = 3 cos 3x
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07 · CONCLUSION
C o nclusio n
1
2
3
A limit shows where a quantity
is heading: for a sequence n →
∞, for a function x → x₀.
Heine’s definition connects the
two ideas: the limit of a function
is expressed through limits of
sequences.
The derivative is a limit of a ratio
of increments. Every formula in
the table of derivatives comes
from it.
Limit of a Sequence. Limit of a Function
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15.

SOURCES
References
1. Stewart J. Calculus: Early Transcendentals. — Cengage Learning.
2. Fichtenholz G. M. A Course of Differential and Integral Calculus. Vol. 1.
3. Demidovich B. P. Problems in Mathematical Analysis.
4. [Textbook or lecture notes recommended by the teacher]
Thank you for your attention!