A limit is the behavior of a function as its inputs approach arbitrarily close to a given value.
Limits are written in the following form:
The expression above states that when approaches arbitrarily close to , the function becomes arbitrarily close to the value , which is called the limit.
We can examine the limit of a simple continuous function, .
The limit at x=0, would simply be f(x)=0, since as the function is continuous and differentiable at every point, .
Indeed for this function, . But this is a special case, in the majority of limits cannot be solved in this manner.
For example, given
(as pictured below)
The limit of as x approaches 0 is 0 (just as in ), but ; is not continuous at (as shown on the right).
In other cases a limit can fail to exist, as approaching the limit from different sides produces conflicting values.
Rigorous Definition of a Limit
This definition is more appropriate for 2nd year calculus students and higher.
The original statement now means that given any , a exists such that if , then .
We use the definition as follows to prove for the function ,
Given any , we choose a such that
If , we can derive .
Properties of a Limit
The limit of a sum of two functions is equal to the sum of the limits of the functions.
The limit of a difference between two functions is equal to the difference between the limits of the functions.
The limit of a product of two functions is equal to the product of the limits of the functions.
The limit of a quotient of two functions is equal to the quotient of the limits of the functions (assuming a non-zero denominator).