In this post I am going to introduce the formal definition of a limit. I am still trying to figure out the symbol formatting so bear with me. If becomes arbitrarily close to a single number as approaches from either side, then the limit of as approaches is , which can be written as . This definition may seem formal but isn’t because exact meanings have not yet been given to the two phrases “ becomes arbitrarily close to ” and “ approaches “. In the figure below we can let represent a (small) positive number. Then the phrase becomes arbitrarily close to means that lies in the interval . Using absolute value, we can write this as . Similarly, the phrase “ approaches ” means that there exists a positive number such that lies in either the interval or the interval . This fact can be concisely expressed by the double inequality . The first inequality states that the distance between and is more than 0, and expresses the fact that . The second inequality says that is within units of .
This brings us to a formal definition which can be stated as following: Let be a function defined on an open interval containing (except possibly at ) and let be a real number. The statement means that for each there exists a such that if , then . This can be expressed symbolically as .