Epsilon-Delta Proofs in Calculus

The precise definition of a limit is given. Epsilon-delta proofs are given which prove that the limit of a linear function is the value of the function at the point. Similarly for a quadratic function. The quadratic function proof involves a subtlety which is an important part of the calculus.

Definition (Limit of a Function) Suppose that the domain of f contains points x arbitrarily close to c but different from c. Then means, for all there exists , such that

for any in the domain of

Concisely, a limit is used to describe the behavior of a function near a point but not at the point. The function need not even be defined at the point. If it is defined there, the value of the function at the point does not affect the limit. Intuitively, means we can make as close to as we wish by taking any sufficiently close to, but different from

Example (Epsilon Delta Proof Linear Functions) Use an epsilon-delta argument to show that

Solution. Let be any real number greater than zero. We need to find a so that,

To do so note that, We find that is acceptable because, if then

Example (Epsilon Delta Proof Quadratic Functions) Use an epsilon-delta argument to show that

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Solution. Let be any real number greater than zero. We need to find a such that,

To do so note that,

which is useful because is small, say and thus,

We find that

is acceptable because, if then

as required.