A number is the limit of a function as approaches if and only if for all numbers there exists a number such that
In other words, given a number we must construct a number such that assuming
we can prove
moreover, this proof must work for all values of : .
Note: this definition is not constructive -- it does not tell you how to find the limit , only how to check whether a particular value is indeed the limit. We use the informal definition of the limit, experience with similar problems, or theorems (L'Hopital's rule, for example), to determine the value, and then can prove the correctness of this value using the formal definition.
Example 1: Suppose we want to find the limit of as approaches . We know that the limit is 9+5=14, and desire to prove this.
We choose (this will be explained later). Then, since we assume
we can show
which is what we wanted to prove.
We chose δ by working backwards from the formula we are trying to prove: . In this case, we desire to prove
so the easiest way to prove it is by choosing . This example, however, is too easy to adequately explain how to choose in general. Lets try something harder:
Example 2: Prove that the limit of as approaches is .
We want to prove that
We choose by working backwards. First, we need to rewrite the equation we want to prove using instead of :
Note: we used the fact that , which can be proven with the triangle inequality.
Word of caution: the above series of equations is not a logical series of steps, and is not part of any proof, but is an informal technique used to help write the proof. We will select a value of so that the last equation is true, and then use the last equation to prove the equations above it in turn (which is what was meant earlier by working backwards).
Note: in the equations above, when was substituted for , the sign was replaced with . This can be done (but is not necessary) because we are not told that , but rather . The justification for this becomes clear when the above equations are used in backwards order in the proof.
We can solve this last equation for using the quadratic formula:
Note: is almost always in terms of . A constant value of (e.g., ) will not work unless the limit is of a constant function (for instance, ).
Now, we have a value of , and we can do our proof:
Here a few more examples of choosing ; try to figure them out before reading the explanation.
Example 3: Prove that the limit of as approaches is .
Example 4: Prove that has no limit as approaches .
Example 5: Prove that
Solution: To do it, we'll look at two cases: and . The case is easy. First let's let . That means we want the values chosen in the domain to map to in the range. We want a delta such that so let's choose . The chosen defines the interval in our domain. This gets mapped to in our range, which is contained in . Notice that doesn't depend on . So for , we widen the interval in the range that we are allowed to map onto, but our interval in the domain stays fixed and always maps to the same sub-interval in the range. So works for any .
Now suppose . We want a such that whenever . So let's assume and work backwards to find a suitable :
Since , we have . Since both numbers above are positive, we can take the (positive) square root of both extremes of the inequality:
The above equation represents the distance, either negative or positive, that can vary from 2 and still be within of 4. We want to choose the smaller of the two extremes to construct our interval. It turns out that for , so choose . As a sanity check, let's try with .