Limit of Functions



AL Pure Mathematics Informal Summary

Limit of Functions 1

Differentiation 2

Applications of Differentiation 4

Indefinite Integral 6

Definite Integral 10

Applications of Definite Integrals 13

Limit of Sequences 14

Binomial Theorem 17

Polynomials 18

Inequalities 20

Complex Numbers 21

System of Linear Equations 27

Coordinate Geometry 29

Limit of Functions

• Two Important Limits

1. [pic] (also, [pic])

2. [pic]

Sandwich Theorem:

If [pic] and [pic] then [pic].

• Continuity

[pic] is continuous at [pic] [pic] [pic]

Differentiation

• Definition (First Principle)

[pic]

• Formula of Differentiation

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

• Rules of Differentiation

1. Product Rule: [pic]

2. Quotient Rule: [pic]

3. Chain Rule: [pic]

e.g. [pic]

• Logarithmic Differentiation

Take ln before differentiate:

e.g. [pic] e.g. [pic]

[pic] [pic]

• Leibniz’s Theorem

[pic]

OR

[pic]

where [pic], etc

• Differentiability

[pic] is differentiable at [pic] if

[pic]

Note: differentiable [pic] continuous

not continuous [pic] not differentiable

Applications of Differentiation

• L’Hospital’s Rule

|For [pic] |For [pic], etc, take ln first |

|e.g. [pic] |e.g. [pic] |

|[pic] |[pic] |

|e.g. [pic] | |

|[pic] | |

• Proving Inequalities

To prove [pic], let [pic] and find [pic]

e.g. Prove [pic] for all [pic].

Let [pic]

[pic]

When [pic], [pic]

|[pic] | |1 | |

|[pic] |--------- |0 |++++++ |

[pic] [pic] is max. at [pic]

[pic] [pic] for all [pic]

[pic]

• Curve Sketching

Steps:

(1) Find [pic] and [pic]

─ fully simplify and factorize them

─ take ln may be useful

(2) Draw table

|e.g. |[pic] | |3 | |

| |[pic] |-------------- |0 |++++++++++ |

| |[pic] |+++++++++++++++++++++++ |

| |[pic] |[pic] |2 |[pic] |

| |[pic] |[pic] |

| |increasing |decreasing |

|[pic] |[pic] |[pic] |

|Concave upward | | |

|[pic] |[pic] |[pic] |

|Concave downward | | |

(3) identify vertical asymptote(s) [pic]

find oblique asymptote(s) [pic] by [pic]

(4) Draw and label asymptotes/ extreme points/point of inflexions/ intercepts

Then draw the curve

Indefinite Integral

• Definition:

If [pic] then [pic]

• Integration Formula

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

Rules of Partial Fractions

e.g.

(1) [pic] (Linear factor)

(2) [pic] (Quadratic factor)

(3) [pic] (Repeated linear factor)

(4) [pic] (Repeated quadratic factor)

• Method of Substitution

e.g. [pic]

[pic] OR

• Integration of Rational Functions ([pic])

─ do long division if [pic]

─ then use partial fractions

• Integration of Rational Functions of [pic]

[pic] [pic]

• Integration of Irrational Functions

For[pic], try to put [pic]

For[pic], try to put [pic]

For[pic], try to put [pic]

• Integration by Parts

[pic]

“derivative transfer”

[pic] [pic] (by parts twice)

• Reduction Formula

─ usually generated by Integration by Parts

[pic]

Definite Integral

• Second Fundamental Theorem

If [pic]

• Method of Substitution

[pic]

• Integration by Parts

[pic]

• Reduction Formula

[pic]

[pic]

When n is odd, When n is even,

[pic] [pic]

• First Fundamental Theorem

[pic]

• Sum an Infinite Series by Definite Integrals

[pic]

e.g.

[pic]

[pic]

• Inequalities on Definite integrals

1. lf f(x)[pic]g(x) [pic][pic]

2. [pic]

Applications of Definite Integrals

• Plane Area

• Volume: Disc Method

• Volume: Shell Method

Limit of Sequences

• Sandwich Theorem

If [pic] for all [pic] , and [pic] then [pic].

e.g. Prove [pic].

Let [pic]

[pic]

[pic]

[pic] [pic]

[pic]

[pic]

[pic]

[pic]

• Monotone Convergence Theorem:

Monotonic increasing + bounded above [pic] Convergent

[pic] [pic]

Monotonic decreasing + bounded below [pic] Convergent

[pic] [pic]

usually proved by Method of Difference or M.I.

e.g. Let [pic] be a sequence of positive numbers, where

[pic] and [pic]

a) Prove that [pic] for all positive integers n.

Try Method of Difference:

[pic]

[pic]

[pic]

[pic] need to have [pic]

[pic] try M.I.

When n = 1, [pic]

[pic][pic] is true for n = 1

Assume [pic]

[pic]

=[pic]

=[pic]

[pic]

[pic][pic] is true for n = k+1

By M.I., [pic]

b) Prove that [pic] exists.

Need to prove monotonic increasing,

try method of difference:

[pic]

=[pic]

=[pic]

=[pic]

[pic] ([pic]

[pic][pic] is monotonic increasing and, by(a), bounded above by 3

[pic][pic] is convergent

c) Find [pic].

Let [pic]

[pic]

[pic]

[pic]

[pic] or [pic]

[pic]

[pic][pic]

[pic]

Binomial Theorem

[pic] total (n+1) terms

[pic] [pic]

[pic]

[pic]

Differentiation and Integration may help you find properties of Binomial coefficients.

e.g. [pic]

Put x = 1, [pic]

e.g. [pic]

Put x = 0, [pic]

Put x = 1, [pic]

[pic]

Polynomials

• Division Algorithm

[pic] Remainder

Quotient

deg [[pic]] < deg [[pic]]

or [pic] (that is, [pic] divisible by [pic])

• Remainder Theorem

When a polynomial [pic] is divided by [pic], the remainder is [pic].

• Factor Theorem

When [pic], [pic] is a factor of the polynomial [pic].

Generally, if [pic] are the roots of a polynomial [pic] of degree [pic],

then [pic], where [pic] is the leading coefficient.

• Euclidean Algorithm (輾轉相除法)

G.C.D. of [pic] and [pic] = G.C.D. of [pic] and [pic]

• Complex Roots Theorem

Let [pic] be a polynomial with real coefficients.

If [pic] is a root of [pic] ([pic]), [pic] is also a root of [pic].

Implications:

1. A polynomial of degree[pic] can be factorized into real linear and/or quadratic factors.

2. A polynomial of odd degree has at least one real root.

• Multiple Roots

[pic] has a multiple root [pic]

[pic]

[pic] and [pic]

• Relation between Roots and Coefficients

Let [pic] be the roots of [pic]

[pic]

Tips: [pic]

[pic]

• Transformations of Polynomial Equations

| |Old Roots |New Roots |Put |

|e.g. |[pic] |[pic] |[pic] |

|e.g. |[pic] |[pic] |[pic] |

|e.g. |[pic] |[pic] |[pic] |

| | |[pic] | |

| | |[pic] | |

| | |[pic] | |

| | |[pic] | |

Inequalities

• Triangle Inequality

[pic]

• Inequalities involving absolute values

[pic]

• A.M.[pic]G.M.

[pic]

OR [pic]

• Cauchy – Schwarz’s Inequality

[pic]

OR [pic]

Proof: [pic] [pic]

[pic]

If [pic] the result is trivial.

If [pic]

[pic]

[pic]

Strategies to prove inequalities:

1. By Method of Difference/ Ratio

2. M.I.

3. By Differentiation

4. Use known results

Complex Numbers

• Definitions: [pic] ([pic])

Let [pic] ([pic])

1. [pic], [pic] Imaginary part is not imaginary!

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

• Polar Form

[pic] ([pic])

or

[pic]

e.g. [pic]

Let [pic], [pic]

[pic] [pic], [pic]

[pic] [pic], [pic]

• Geometric Relationship

[pic] is purely imaginary [pic] [pic] [pic] [pic]

[pic] is purely imaginary

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

• Locus Problems

[pic] [pic]

General method: Put [pic]

• DeMoivre’s Theorem

[pic] (n is any integers)

Roots of [pic]:

[pic] [pic]

e.g. Solve [pic]

[pic]

Hence we can factorize [pic] into real linear/quadratic factors

[pic]

Matrices & Determinants

• Arithmetic of Matrices

Addition/Subtraction: [pic]

Scalar Multiplication: [pic]

Multiplication: [pic]

Transpose: [pic], [pic]

• Determinants

[pic]

[pic]

( expanded along 1st row )

* Can be expanded along any row/ column [pic]

Some properties:

1. [pic] Interchanging two rows gives negative sign

2. [pic] Common factor extracted from a row/ column

3.

4. [pic]

[pic]

5. [pic]

• Inverse

Definition: [pic]

Existence: [pic]

[pic]

[pic]

[pic]

• Applications in Cooradinate Transformation

“output” [pic] “input”

transformation matrix

Rotation: [pic]

Reflection: [pic]

Enlargement: [pic]

System of Linear Equations

• Notation

[pic]

[pic]

• Finding Unique Solution by Cramer’s Rule [pic]

[pic]

[pic]

• [pic] & Consistency

| |[pic] |[pic] |

|[pic] |unique solution |unique trivial solution |

| |obtained by |(0, 0, 0) |

| |Cramer’s Rule | |

| |Method of Inverse Matrix | |

| |G. E. | |

|[pic] |no solution (inconsistent) |infinitely many solutions |

| | |(obtained by G. E.) |

| |infinitely many solutions | |

| |(obtained by G. E.) | |

• Gaussian Elimination

Goal: [pic]

Coordinate Geometry

• Angle between two straight lines

[pic]

• Distance from a point to a line

[pic]

• Equations of straight lines

Point-slope form: [pic]

Slope-intercept form: [pic]

• Equations of Circles

Standard form: [pic]

General form [pic]

Centre [pic]

Radius [pic]

• Equations of Parabola

• Equations of Ellipses

[pic]

• Equations of Hyperbola

[pic] [pic]

• Family of Straight Lines / Circles

[pic]

• Locus Problems

Let the movable point / variable point interested be (x, y).

Try to find an equation connecting x and y.

• Equations of Tangents and Chords

[pic][pic] [pic] [pic] [pic]

[pic]

[pic][pic] [pic] [pic] [pic]

-----------------------

Put [pic]

lf [pic] for all n, then [pic].

lf [pic] for all n, then [pic].

[pic]

includes values of x

(1) at which [pic] and [pic]

(2) at which [pic] [pic] or [pic] is undefined

(3) the expression inside absolute value sign is zero

e.g. [pic]

include [pic]

[pic]

completing square

[pic]

[pic]

[pic]

[pic]

Put u = x + 1

du = dx

|x |2 |1 |

|u |3 |2 |

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Note: In general,

[pic]

[pic]

[pic]

Tips:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

partial fractions

[pic] use summation notation

[pic]

[pic]

[pic]

[pic]

[pic]

e.g.

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

e.g.

e.g.

[pic] ‘make’ [pic]

[pic] ‘make’ [pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

principal argument

[pic]

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