There are two popular ways to write that the limit of a sequence {a_n} is L. This is the simpler way:

a_n→L.

It is very convenient and it allows us to make algebraic changes before evaluating the limit, for instance like this.

3n − n = 2n→∞.

Note that the equality means algebraic equality of expressions. The notation therefore says that we have two expressions that are equal, we can evaluate the limit of the second one, and since the expressions are the same, the limit is valid also for the first expression. This is also the limitation of this notation. Some methods require changes that do not use equality of expressions, most notably the l'Hospital rule which we cannot write using this simple way. This brings us to the second notation.

Standard notation for limit is

Sometimes we skip the caption under "lim", since n cannot go anywhere else but to infinity. Recall that the symbol "lim(a_n)" represents a number (the outcome of the limit here called L). This means that when we form an equality "lim( ) = lim( )", then it specifies eqality of the two numbers represented by the limits, in other words, we are allowed to change the expression inside "lim" quite a bit as long as the limit stays the same. Thus we can do more than with the previous notation, for instance use l'Hospital's rule. Another advantage compared to the previous notation is the possibility to split the limit into several simpler limits using rules and then evaluate some sooner than others, therefore we often use this notation for more complex calculations.

People sometimes make a mistake by dropping the symbol "lim" too early or keeping it for too long. In fact it is quite simple. The symbol "lim" represents the answer to the question "what is the limit", so we stop writing it exactly when we supply an answer, and we have to keep it as long as we work with an expression that features the variable. Note that the variable must disappear from the answer (the outcome of a limit is a number). Thus

lim(3n − n) = 2n

is wrong,

lim(3n − n) = lim(∞)

is also wrong, but this is correct:

lim(3n − n) = lim(2n) = ∞.