Indeterminate form

In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an indeterminate form. The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century.

The indeterminate forms typically considered in the literature are denoted 0/0, ∞/∞, 0 × ∞, ∞ − ∞, 0⁰, 1^∞ and ∞⁰.

Discussion

The most common example of an indeterminate form occurs as the ratio of two functions, in which both of these functions tend to zero in the limit, and is referred to as "the indeterminate form 0/0". As x approaches 0, the ratios x/x³, x/x, and x²/x go to ∞, 1, and 0 respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is 0/0, which is undefined. So, in a manner of speaking, 0/0 can take on the values 0, 1 or ∞, and it is possible to construct similar examples for which the limit is any particular value.

Arsenal F.C.

Arsenal Football Club is a professional football club based in Holloway, London. The club play in the Premier League, the top flight of English football. One of the most decorated clubs in English football, they have won 41 domestic and two European trophies, including 13 First Division and Premier League titles and a record 12 FA Cups.

Arsenal were founded in 1886 in Woolwich and in 1893 became the first club from the south of England to join The Football League. In 1913, they moved north across the city to Arsenal Stadium in Highbury. Herbert Chapman changed their traditional red and white kit in 1933 to include white sleeves. In 2006, they moved to the Emirates Stadium in nearby Holloway. Arsenal contest the North London derby with rivals Tottenham Hotspur.

They have never left the top flight since 1919 – the ongoing longest run in English Football. This has helped them accumulate the second most top flight wins and points, and finish the 20th century with the highest average league position. In the 1930s, they won five League Championships and two FA Cups, and another FA Cup and two Championships after the war. In 1970–71, they won their first League and FA Cup Double. George Graham and Arsène Wenger then oversaw five League titles, seven FA Cups, two more Doubles, and a Champions League Final. Under Wenger, Arsenal also set the current top flight records for longest win streak and longest unbeaten streak, and completed a 38 match season unbeaten, surpassing Preston North End from 115 years before.

Football team

A football team is the collective name given to a group of players selected together in the various team sports known as football.

Such teams could be selected to play in a match against an opposing team, to represent a football club, group, state or nation, an All-star team or even selected as a hypothetical team (such as a Dream Team or Team of the Century) and never play an actual match.

There are several varieties of football, notably Association football, Gridiron football, Australian rules football, Gaelic football, rugby league, and rugby union. The number of players selected for each team within these varieties and their associated codes can vary substantially. In some, use of the word "team" is sometimes limited to those who play on the field in a match and does not always include other players who may take part as replacements or emergency players. "Football squad" may be used to be inclusive of these support and reserve players.

The term football club is the most commonly used for a sports club which is an organised or incorporated body with a president, committee and a set of rules responsible for ensuring the continued playing existence of one or more teams which are selected for regular competition play (and which may participate in several different divisions or leagues). The oldest football clubs date back to the early 19th century. The words team and club are sometimes used interchangeably by supporters, although they typically refer to the team within the club playing in the highest division or competition.