Cambridge IGCSE Mathematics Syllabus

Cambridge IGCSE Mathematics

Syllabus code 0580

Cambridge IGCSE Mathematics (with coursework)  

Syllabus code 0581

For examination in June and November 2012

IGCSE Syllabus (Click) 

About 30 years ago they were separate exams. The GCE O Level (General Certificate of Education "Ordinary" Level) was a direct precident to the GCE A (Advanced) Level exam. That is, you had to pass the O Level in a subject before being allowed to sit the A Level in the same subject.

GCE O Levels were purely examination based. The pupil had to study a (typically) 2 year course and the exam would be based on the whole syllabus. The exam was "sudden death" in that there was a Pass or Fail. You could get grades of Pass from A to C but these must not be equated to today's GCSE grades A to C because of the way the system was structured.

CSE (Certificate of Secondary Education) was run in parallel to the GCE O Level courses. It was intended for those who were academically poor at exams but were good at coursework. The grade of CSE was based on part coursework and part exam.
It was generally thought of an inferior in value to GCE O Level. This was partly because the percentage required before you were "failed" was inreadibly low. Therefore it was considered that a Grade A CSE was equivalent to a Grade C Pass at O level.
This gave many bright CSE pupils (they were not called students until they entered college in those days, the modern useage is an Americanism) who were bad at "memorising" a chance to study for A Level.

As a result of Political Correctness the Labour governments of the 1970's abolished the separate curriculums and merged them into the General Certificate of Secondary Education (GCSE).

They tried to abolish the GCE A Level too (it was called "elitist") but failed because it is used by Universities as an admission guide. Because universities have to have some means of determining who can join their courses (degrees are academically demanding - they have to be or they'd be worthless), the exam based A Level cannot be beaten.
The A Level exams are more than just a memory test. To get a good grade you have to demonstrate a thorough understanding of the subject, not just regurgitate memorised facts.
To be fairer to people who thrive on coursework but "freeze" in exams the AS Level was introduced which provides a safety net. This enables you to gain "points" which can count towards university admission.

Many have tried to find an alternative to A Levels but the only one that comes close is the International Baccalauriate, and that is one heck of a doozie of a course!

GCE 'O'-level is still available and is an alternative to the GCSE exam which was introduced in the late 1980's to replace the CSE (certificate of secondary education) which was a much easier exam than GCE.

Oxford University will advise as to whether your 'intermediate' exams, taken in Pakistan, would be deemed as an acceptable equivalent to GCSE/GCE 'O'-level (the 'O' stands for 'ordinary' by the way; the 'A' in 'A'-level = 'Advanced'

You would be able to take one exam in the subject of your choice (mathematics) should you need to take it. If you go to the websites I've quoted below, you will be able to download past papers.

If you need further help/advice, feel free to e-mail me.
Check AQA and Edexcel website for more information.

Source(s):

http://www.edexcel-international.org/hom…

http://www.aqa.org.uk/

 

 

http://www.aqa.org.uk/over/qual/gceas.ph…
http://www.edexcel-international.org/qua…

 

 

 

diff between exams of IGCSE and GCSE/

However, from what Ive heard, GCEs are harder... I personally took the GCEs and while studying we often used GCSE books and found the examinations and tets in it to be a piece of cake... Personally, I dont even think the GCEs are tough, if youre consistent... I did the same and I got straight As in both O and A levels... I also believe that solving and analyzing the past 10 years examinations helps a great deal... I saw lots of Multiple Choice Questions both in O and A levels that were repeated...
www.ucles.org
(Site for the university of cambridge local examinations syndicate)

www.cie.org.pk
(Site for Cambridge International Examinations, where u can find downloadable copies of the syllabi)

The best thing about Alevels in that you get credit at university level... I got 24 credits for my alevels, and that like more than what we take in one semester at college...



I'm surprised that you think otherwise. When I took O-Levels we were always told that GCSE was harder. And the topic came up a lot too because our college had shifted their affiliations from GCSE at Cambridge to GCE at London. Either way all Levels' systems are quite fantastic and are much better than FSc and Matric.

You are right about the bit that to get good grades in any Levels exam, do lots and lots of old exams. Frankly I found doing old exams more useful than actually doing course coverage. So people who are doing O/A Levels now, do lots of old exams and the A's will be flooding in.

 The fact that IGCSEs have a grade higher than A (the A*) makes Grade A a little ummm... below par.




 GCSE's are the General Certificate in Secondary Education in the Uk- equivalent of the " o" Levels - the IGCSE's are the International Version that are studied globally


 They replaced the O level and were introduced in 1990 afer the Uk educational system was " Reorganised".....they are more modular and therefore coursework weighted.


also note that GCE's (are a-levels ) are studied after GCSE in uk. Thus GCE are meant to be hard than GCSE's

Cambridge IGCSE

Overview

Cambridge IGCSE is the world’s most popular international curriculum for 14-16 year olds, leading to globally recognised and valued Cambridge IGCSE qualifications. It is part of the Cambridge Secondary 2 stage.

Schools worldwide have helped develop Cambridge IGCSE, which provides excellent preparation for the Cambridge Advanced stage including Cambridge International AS and A Levels and Cambridge Pre-U, as well as other progression routes. It incorporates the best in international education for learners at this level. It develops in line with changing needs, and is regularly updated and extended. Cambridge IGCSE teachers can draw on excellent resources, training and advice from subject experts.

Building a curriculum

Cambridge IGCSE encourages learner-centred and enquiry-based approaches to learning. It develops learners’ skills in creative thinking, enquiry and problem solving, giving learners excellent preparation for the next stage in their education. Schools can build a core curriculum, extend it to suit their learners and introduce cross-curricular perspectives. Clearly defined learning outcomes and content, mean Cambridge IGCSE is compatible with other curricula and is internationally relevant and sensitive to different needs and cultures.

Schools can offer any combination of subjects. Each subject is certificated separately. Over 70 subjects are available, including more than 30 language courses, offering a variety of routes for learners of different abilities. Cambridge IGCSE develops learner knowledge, understanding and skills in:

• Subject content
• Applying knowledge and understanding to familiar and new situations
• Intellectual enquiry
• Flexibility and responsiveness to change
• Working and communicating in English
• Influencing outcomes
• Cultural awareness

Assessment

Assessment for Cambridge IGCSE includes written and oral tests, coursework and practical assessment. Schools have the option of assessing learners using only external examinations or, in most subjects, combining examinations with coursework. Coursework is set and marked by the teacher and externally moderated by Cambridge.

Teachers who have received training from Cambridge, or who possess suitable experience of marking coursework may carry out this assessment. In most subjects there is a choice between core and extended curricula, making IGCSE suitable for a wide range of abilities. Each learner’s performance is benchmarked using eight internationally recognised grades. There are clear guidelines which explain the standard of achievement for different grades. Cambridge IGCSE examination sessions occur twice a year, in May/June and October/November. Results are issued in August and January.

Cambridge progression

Cambridge offers routes candidates can follow from post-kindergarten stage through to university entrance. Cambridge's provision also includes first-class support for teachers through publications, online resources, training, workshops and professional development.

University Cambrige 

IGCSE Mathematics (FAQ)

IGCSE Mathematics (0580)

IGCSE Mathematics (with coursework) (0581) 

Frequently Asked Questions

READ

Differences between IGCSE and O' Level Mathematics

 The main differences between O Level and IGCSE Mathematics are as follows:

• IGCSE Maths is available with or without Coursework, whereas O Level Maths is only available without Coursework. 
• IGCSE has Core and Extended options (grades C-G and A* to E available respectively), O level has grades A to E available. 
• O level Maths has a non-calculator paper whereas IGCSE Maths requires a calculator for both papers.
• Examining time: 4.5 hours for O' level Maths compared with 3 hours for Core IGCSE Maths and 4 hours for Extended IGCSE Maths. 
• There is no question choice for IGCSE Maths; for O level Maths there is limited question choice in Paper 2. 
• O level Maths question paper weighting is 50% per paper; IGCSE Maths is 35% for the first paper, 65% for the second.

Ratio

Ratio

A ratio is a comparison of two numbers. 

Examples:

Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange.

1) What is the ratio of books to marbles?

A ratio is a comparison between two or more like quantities in the same units.

Note:

The ratio 1 : 2 is read as '1 to 2' or '1 is to 2'.

Scale Factor

If the ratio is expressed in the form 1 : n, then n is called the scale factor.
E.g.  5 : 20 = 1 : 4
So, 4 is the scale factor.

Note:

We can use ratios to compare more than two quantities conveniently.  Fractions are not usually suitable for this.

 Ratios are used in areas including concentration of solutions, drug dosages, financial mathematics and gears.

Comparing Ratios

To compare ratios, write them as fractions. The ratios are equal if they are equal when written as fractions.
Example:
Are the ratios 3 to 4 and 6:8 equal?
The ratios are equal if 3/4 = 6/8.
These are equal if their cross products are equal; that is, if 3 × 8 = 4 × 6. Since both of these products equal 24, the answer is yes, the ratios are equal.

proportion (Meaning)

pro·por·tion  (pr-pôrshn, -pr-)

n.

1. A part considered in relation to the whole.

2. A relationship between things or parts of things with respect to comparative magnitude, quantity, or degree: the proper proportion between oil and vinegar in the dressing.

3. A relationship between quantities such that if one varies then another varies in a manner dependent on the first: "We do not always find visible happiness in proportion to visible virtue" (Samuel Johnson).

4. Agreeable or harmonious relation of parts within a whole; balance or symmetry.

5. Dimensions; size. Often used in the plural.

6. Mathematics A statement of equality between two ratios. Four quantities, a, b, c, d, are said to be in proportion if ^a/_b = ^c/_d .

tr.v. pro·por·tioned, pro·por·tion·ing, pro·por·tions

1. To adjust so that proper relations between parts are attained.

2. To form the parts of with balance or symmetry.

Proportion

A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. 

When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion.

We compare rates just as we compare ratios, by cross multiplying

Rate

A rate is a ratio

Problems involving rates typically involve setting two ratios equal to each other and solving for an unknown quantity, that is, solving a proportion.


When comparing rates, always check to see which units of measurement are being used. For instance, 3 kilometers per hour is very different from 3 meters per hour! 

Important:

One of the most useful tips in solving any math or science problem is to always write out the units when multiplying, dividing, or converting from one unit to another.

Average Rate of Speed

The average rate of speed for a trip is the total distance traveled divided by the total time of the trip.

Increasing or Decreasing a Quantity in a Given Ratio

If the ratio of a new quantity to an old quantity can be expressed as an improper fraction, then the new quantity is greater than the old quantity.  Applying this ratio to the old quantity is known as increasing the old quantity in a given ratio. If the ratio of a new quantity to an old quantity can be expressed as a proper fraction, then the new quantity is less than the old quantity.  Applying this ratio to the old quantity is known as decreasing the old quantity in a given ratio.

Example 8

Increase 20 in the ratio 3 : 2.

Solution:

Example 9

Decrease 32 in the ratio 3 : 4.

Using a Ratio to Solve Problems

If the ratio of two quantities is known and one of the quantities is given, then the other quantity can be calculated. Example 10 A company wants to reduce its operating cost in the ratio 2 : 3. If the operating cost was $51000 last year, what would be its target cost? to be continue............. http://www.mathsteacher.com.au/year8/ch06_ratios/05_divide/ratio.htm Golden_ratio

Percentages (%)

When you say "Percent" you are really saying "per 100"

And 25% means 25 per 100

Formula for percentage

Examples #1:

25 % of 200 is____ 

In this problem, of = 200, is = ?, and % = 25

We get:

is/200 = 25/100

Since is in an unknown, you can replace it by y to make the problem more familiar

y/200 = 25/100

Cross multiply to get y × 100 = 200 × 25

y × 100 = 5000

Divide 5000 by 100 to get y

Since 5000/100 = 50, y = 50

So, 25 % of 200 is 50




Now, we will take examples to illustrate how to use the formula for percentage on the right

Examples #4:

To use the other formula that says part and whole, just remember the following:

The number after of is always the whole

The number after is is always the part

If I say 25 % of___ is 60, we know that the whole is missing and part = 60

Your proportion will will like this:

60/whole = 25/100

After cross multiplying, we get:

whole × 25 = 60 × 100

whole × 25 = 6000

Divide 6000 by 25 to get whole

6000/25 = 240, so whole = 240

Therefore, 25 % of 240 is 60














Examples #2:

What number is 2% of 50 ?

This is just another way of saying 2% of 50 is___

So, set up the proportion as example #1




Examples #3:

24% of___ is 36

This time, notice that is = 36, but of is missing

After you set up the formula, you get:

36/of = 24/100

Replace of by y and cross multiply to get:

36/y = 24/100

Percentage increase and decrease

Sometimes due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the initial value of that quantity. For example, if an item is initially priced at $200 and the price rises 10% (an increase of $20), the new price will be $220. Note that this final price is 110% of the initial price (100% + 10% = 110%).

Some other examples of percent changes:

• An increase of 100% in a quantity means that the final amount is 200% of the initial amount (100% of initial + 100% of increase = 200% of initial); in other words, the quantity has doubled.
• An increase of 800% means the final amount is 9 times the original (100% + 800% = 900% = 9 times as large).
• A decrease of 60% means the final amount is 40% of the original (100% − 60% = 40%).
• A decrease of 100% means the final amount is zero (100% − 100% = 0%).

http://en.wikipedia.org/wiki/Percentage

(New Value / Old Value)  * 100   =  % Value

Percent Increase (U Tube)

http://www.basic-mathematics.com/formula-for-percentage.html

Fractions Made Easy

Fraction

A fraction (from Latin: fractus, "broken") represents a part of a whole or, 

more generally, any number of equal parts. 

When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.

A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists of a numerator and adenominator—the numerator representing a number of equal parts and the denominator indicating how many of those parts make up a whole. 

An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts equal a whole. The picture to the right illustrates 3/4 of a cake.

A cake with one quarter removed. The remaining three quarters are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each quarter of the cake is denoted by the fraction 1/4.

Fractional numbers can also be written without using explicit numerators or denominators, 

by using decimals, 

percent signs, or 

negative exponents (as in 0.01, 1%, and 10⁻² respectively, 

all of which are equivalent to 1/100).

 An integer (e.g. the number 7) has an implied denominator of one, meaning that the number can be expressed as a fraction like 7/1.

Other uses for fractions are to represent ratios and to represent division. 

Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and 

the division 3 ÷ 4 (three divided by four). 

In mathematics the set of all numbers which can be expressed as a fraction m/n, where m and n are integers and n is not zero, is called the set of rational numbers and is represented by the symbol Q. 

The word fraction is also used to describe continued fractions,algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2  and π/4 

http://en.wikipedia.org/wiki/Fraction_(mathematics)

Forms of fractions

A common fraction (also known as a vulgar fraction or simple fraction) is a rational number written as an ordered pair of integers, called the numerator and denominator, separated by a line. 

Proper and improper common fractions

A common fraction is said to be a proper fraction if the absolute value of the numerator is less than the absolute value of the denominator—that is, if the absolute value of the entire fraction is less than 1. A vulgar fraction is said to be an improper fraction (U.S., British or Australian) or top-heavy fraction (British, occasionally North America) if the absolute value of the numerator is greater than or equal to the absolute value of the denominator (e.g. )

Mixed numbers

A mixed numeral (often called a mixed number, also called a mixed fraction) is the sum of a whole number and a proper fraction. This sum is implied without the use of any visible operator such as "+". For example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other: .

This is not to be confused with the algebra rule of implied multiplication. When two algebraic expressions are written next to each other, the operation of multiplication is said to be "understood". In algebra,  for example is not a mixed number. Instead, multiplication is understood: .

Complex fractions

In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number

Compound fractions

Fractions Made Easy

 All it takes is to look at it in terms of everyday objects, like pizza!

Table of Contents

1. The Basics
2. Numerators and Denominators
3. Kinds of Fractions
4. Type of Fractions Quiz
5. Converting between Mixed and Improper Forms
6. Comparing fractions
7. Equivalent Fractions
8. Comparing Fractions Quiz!
9. Least Common Denominators and Reciprocals
10. LCD and Reciprocals Quiz!
11. Adding and Subtracting Fractions
12. Multiplying and Dividing Fractions
13. A little fraction arithmetic quiz!
14. Lowest Terms
15. Lowest Terms Quiz
16. Fractions stuff on Amazon
17. Love 'em or hate 'em.
18. Other Fun Math Resources!
19. Comments, suggestions, outpourings of emotion

 http://www.squidoo.com/fractions-made-easy

 

 

Vector Spaces

 Vector Spaces

Vector spaces are the subject of linear algebra. Today, vector spaces are applied throughout mathematics, science and engineering.

 A vector space is nothing more than a collection of vectors that satisfies a set of axioms.

 vector space origin

http://www.math.montana.edu/frankw/ccp/multiworld/virtual/Vectors1/learn.htm 

Mathematics Contents 9 (City School)

Mathematics Contents  9 (City School)

1. Direct & Indirect Proportions                          

     D-2;  

     Ex. 2b, 2c, 2e, 2f

2. Algebraic Multiplication and Formulae

     D-2,

     Ex. 4f, 

     Q3

     4j, 4k

     Q 1 - 5

3. Indices & Standard Form

    D-3, 

    ????; 2e

4. Pythagorus Theorem

   D - 2

  Ex 6a; 6b

  Q 1 - 16

5. Trigonometical Ratios

  D-3

  Ex 6a; 6b, 6c; 6d; 6e; 6f; 6g

  Q 1 - 10

6. Volume & Surface

   D -2 

   Ex. 7b; 7c

7. Mensuration, Arc Length, Surface Area, ??? Radian

    D -3 

   Ex. 12a, 12b

8. Sets

  D-2

Ex. 10a; 10b, 10c, 10d

LIMITS

 

CENTRAL TO CALCULUS is the value of the slope of a line,

, but when

the terms become almost

0 0

.  To evaluate the slope, that rate of change,

under those vanishing conditions, requires the idea of a limit.  And central to the idea of a limit is the idea of a sequence of rational numbers.

A sequence of rational numbers

We encounter such a sequence in geometry when we determine a value for π, which is the ratio of the circumference of a circle to the diameter.  To do that, we inscribe in the circle a regular polygon. The ratio of the perimeter of the polygon to the diameter, which we can actually calculate, will be an approximation to π.  And as we increase the number of sides -- that is, if we consider a sequence of polygons:  60 sides, 61 sides, 62, 63, 64, and so on -- then the sequence of those ratios gets closer and closer to π.  Now, the circle is never equal to any polygon. But by considering a sufficiently large number of sides, the difference between the circle and that polygon -- the error -- will be less than a certain small number that we specify.  Less, say, than

0.00000000000000000000000000000001

That is the idea of a sequence approaching a limit, or a boundary.
By that process we can approximate the value of π, which is the limit
of the ratio of perimeters to diameter, as closely as we care to.

Problem 1.    The student surely can recognize the number that is the limit of this sequence of rational numbers.

3,  3.1,  3.14,  3.141,  3.1415,  3.14159,  3.141592, . . .

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

π

We speak of a sequence as being infinite, which, in analogy with the sequence of natural numbers, is a brief way of saying that, no matter which term we name, there is a rule or a pattern that allows us to name one more.

See The mathematical existence of numbers.

The limit of a variable

Consider this sequence of values of a variable x:

1.9,  1.99,  1.999,  1.9999,  1.99999, . . .

Those values are getting closer and closer to 2 -- they are approaching 2 as their limit.  2 is the smallest number such that no matter which term of that sequence we name, it will be less than 2.

By "closer and closer" we mean the following.  Choose an extremely small positive number.  For example, "1 over the national debt"  Then we can name a term of that sequence such that the absolute value of the difference between it and 2  will be less than that small number -- and the same will be true of any subsequent term that we name.
(We say the absolute value because the terms are less than 2, and so the difference itself will be negative.)
When a variable x approaches a number l as a limit, we symbolize that as  x l.  Read:  "The values of x approach l as a limit," or simply, "x approaches l."  In the example above,  x 2.  "x approaches 2."
We also say that a sequence converges to a limit.  The sequence above converges to 2.

By a sequence in what follows, we mean an ordering of rational numbers according to a rule or an indicated pattern.  Here, for example, is a sequence that approaches 0:

0.1,  0.01,  0.001,  0.0001,  0.00001,  and so on.

Left-hand and right-hand limits

Now the sequence we chose were values less than 2.  Hence we say that x approaches 2 from the left.  We write

x 2−

But we can easily construct a sequence of values of x that converges to 2 from the right; that is, a sequence of values that are more than 2.
For example,

2.2,  2.1,  2.01,  2.001,  2.0001,  2.00001, .  .  .

In this case, we write  x 2+ .

But again, no matter what small number we specify, if we go far enough out in that sequence, the value of a difference |x − 2| will be less than that small number.  And so will all subsequent differences that we might name.

Again, when we say that the values of x "approach a limit l," that limit is never a value of x as it approaches l.  There is always a difference between the values of x and their limit.  The limit is the boundary beyond which no member of the sequence will pass.

We summarize this in the following definition.  But first, x is not the only variable.  y is a variable.  And y will be a function of x -- f(x) -- which is also a variable.  And when we come to the definition of the derivative, Δx or h will be the variable.  In the following, then, we will use the letter v to represent any variable.

DEFINITION 2.1.  The limit of a variable.  We say that a sequence of values of a variable v approaches a limit l (a number which is not a term in the sequence)  if, beginning with a certain term of the squence, the absolute value of  v − l  for that term and any subsequent term we might name  is less than an extremely small positive number that we specify.

When that condition is satisfied, we write  v l.

Thus if a sequence of values of a variable approaches a limit, there is always a difference between the limit and the terms of the sequence.  But the difference becomes less than the specified small number of the definition, which is the error we are willing to tolerate. Therefore, because we specify that small number, we say that we can come as close to that limit as we please.

If Δx, then, is the variable that approaches the limit 0 (as it does when we determine the derivative), then Δx is never equal to 0.

The limit of a function

We have defined the limit of a variable, but what we typically have is a function of a variable -- which is also a variable.  For example,

y = f(x) = x².

Now, a sequence of values of x will force a sequence of values of f(x). The question is:  If the values of x approaches a limit, will the corresponding values of f(x) also approach a limit?  If that is the case -- if f(x) approaches a limit L when x approaches a limit l -- then we write

"The limit of f(x) as x approaches l, is L."

In fact, let us see what happens to  f(x) = x²  as x 2−.  Suppose again that x assumes this sequence of values:

1.9,  1.99,  1.999,  1.9999,  1.99999, . . .

x² will then assume this sequence:

(1.9)²,  (1.99)²),  (1.999)²,  (1.9999)²,  (1.99999)², . . .

It is easy to see that x² approaches 2² = 4.

Again, this means that, beginning with a certain term of the x² sequence, the absolute values of the differences between the terms and 4  will be less than any extremely small positive number that we might specify.

Moreover, if we consider a sequence  x 2+:

2.2,  2.1,  2.01,  2.001,  2.0001,  2.00001, .  .  .

then x² becomes this sequence:

(2.2)²,   (2.1)²,   (2.01)² ,   (2.001)²,   (2.0001)²,  .  .  .

And that sequence also approaches 4.  Therefore 4 is the limit of x² whether x approaches 2 from the right or from the left.  And so we can drop the + or − signs and simply write:

To summarize:

DEFINITION 2.2.  A function "has a limit."  We say that a

function f(x) "has a limit" L as x approaches , if  for every sequence of values of x that approach as a limit (Definition 2.1) -- whether from the left or from the right -- the corresponding values of f(x) approach L as a limit (Definition 2.1).

If that is the case, then we write:

"The limit of f(x) as x approaches l  is  L."

In other words, for the limit of f(x) to exist as x approaches l , the left-hand and right-hand limits must be equal.

if and only if

When we say, then, that a function has a limit, we mean that Definition 2.2 has been satified. In practice, it is not necessary to actually produce the requirements of the definition. The theorems on limits imply them.

The most important limit -- the limit that differential calculus is about -- is called the derivative. All the other limits studied in Calculus I are logical fun and games, never to be heard from again.

Now here is an example of a function that does not approach a limit:

As x approaches 2 from the left,  f(x) approaches 1.  As x approaches 2 from the right,  f(x) approaches 3.  The left- and right-hand limits are not equal.  Therefore,  f(x) does not approach any limit as x approaches 2. Definition 2.2,

http://www.themathpage.com/acalc/limits.htm#sequence