Limit
The notion of Limit is considered one of the most important ideas in Calculus and was the one that took several efforts before it was finally formalised.
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[hide]Definition
Although Limit can be defined in several settings, we will give the definition for an ordinary (real to real) function.
Let
Let be a cluster point of
Let
Let
We say that iff
such that
Intuitive Meaning
The formal definition of a limit given above is not necessarily easy to understand. We can instead offer the following informal explanation: a limit is the value to which the function grows close. For example, , because whenever is close to 2, the function grows close to 4. In this case, the limit of the function is exactly equal to the value of the function. That is, . This is because the function we chose was a continuous function. However, not all functions have this property. For example, consider the function over the reals defined to be 0 if and 1 if . Although the value of the function at 0 is 1, the limit is, in fact, zero. Intuitively, this is because no matter how close we get to zero, as long as we never actually reach zero, will always be close to (in fact equal to) zero. Note that if our definition required only that , the limit of this function would not exist.
Left and Right Hand Limits
Left and right hand limits are the limits taken as a point is approached from the left and from the right, respectively. The left hand limit is denoted as , and the right hand limit is denoted as .
If the left hand and right hand limits at a certain point differ, than the limit does not exist at that point. For example, if we consider the step function (the greatest integer function) , we have , while .
Existence of Limits
Limits do not always exist. For example does not exist, since, in fact, there exists no for which there exists satisfying the definition's conditions, since grows arbitrarily large as approaches 0. However, it is possible for not to exist even when is defined at . For example, consider the Dirichlet function, , defined to be 0 when is irrational, and 1 when is rational. Here, does not exist for any value of . Alternatively, limits can exist where a function is not defined, as for the function defined to be 1, but only for nonzero reals. Here, , since for arbitrarily close to 0, .
A limit exists if the left and right hand side limits exist, and are equal.
Sequential Criterion
Let , Let be a cluster point of , Let and let Let
Then
(1) if and only if
(2) sequence that converges to , the sequence converges to
Other Properties
Let and be real functions. Then:
- given that .
- If a limit exists, it is unique.