The limit of a function at a point in its domain (if it exists) is the value that the function approaches as its argument approaches The concept of a limit is the fundamental concept of calculus and analysis. It is used to define the derivative and the definite integral, and it can also be used to analyze the local behavior of functions near points of interest.
Informally, a function is said to have a limit at if it is possible to make the function arbitrarily close to by choosing values closer and closer to . Note that the actual value at is irrelevant to the value of the limit.
The notation is as follows:
which is read as "the limit of as approaches is "
Main Article: Epsilon-Delta Definition of a Limit
The precise definition of the limit is discussed in the wiki Epsilon-Delta Definition of a Limit.
Formal Definition of a Function Limit:
The limit of as approaches is , i.e.
if, for every , there exists such that, for all ,
In practice, this definition is only used in relatively unusual situations. For many applications, it is easier to use the definition to prove some basic properties of limits and to use those properties to answer straightforward questions involving limits.
The most important properties of limits are the algebraic properties, which say essentially that limits respect algebraic operations:
Suppose that and Then
These can all be proved via application of the epsilon-delta definition. Note that the results are only true if the limits of the individual functions exist: if and do not exist, the limit of their sum (or difference, product, or quotient) might nevertheless exist.
Coupled with the basic limits where is a constant, and the properties can be used to deduce limits involving rational functions:
Let and be polynomials, and suppose Then
Note that is a more difficult case; see the Indeterminate Forms wiki for further discussion.
Let and be positive integers. Find
Immediately substituting does not work, since the denominator evaluates to First, divide top and bottom by to get
Plugging in to the denominator does not give so the limit is this fraction evaluated at which is
It is important to notice that the manipulations in the above example are justified by the fact that is independent of the value of at or whether that value exists. This justifies, for instance, dividing the top and bottom of the fraction by since this is nonzero for
A one-sided limit only considers values of a function that approaches a value from either above or below.
The right-side limit of a function as it approaches is the limit
The left-side limit of a function is
The notation "" indicates that we only consider values of that are less than when evaluating the limit. Likewise, for "" we consider only values greater than . One-sided limits are important when evaluating limits containing absolute values , sign , floor functions , and other piecewise functions.
The image above demonstrates both left- and right-sided limits on a continuous function
Find the left- and right-side limits of the signum function as
Consider the following graph:
From this we see and
Determine the limit
Note that, for can be written as . Hence, the limit is
By definition, a two-sided limit
exists if the one-sided limits and are the same.
Compute the limit
Since the absolute value function is defined in a piecewise manner, we have to consider two limits: and
Start with the limit For So
Let us now consider the left-hand limit
So the two-sided limit does not exist.
The image below is a graph of a function . As shown, it is continuous for all points except and which are its asymptotes. Find all the integer points where the two-sided limit exists.
Since the graph is continuous at all points except and , the two-sided limit exists at , , , and . At there is no finite value for either of the two-sided limits, since the function increases without bound as the -coordinate approaches (but see the next section for a further discussion). The situation is similar for So the points , , , and are all the integers on which two-sided limits are defined.
As seen in the previous section, one way for a limit not to exist is for the one-sided limits to disagree. Another common way for a limit to not exist at a point is for the function to "blow up" near i.e. the function increases without bound. This happens in the above example at where there is a vertical asymptote. This common situation gives rise to the following notation:
Given a function and a real number we say
If the function can be made arbitrarily large by moving sufficiently close to
There are similar definitions for one-sided limits, as well as limits "approaching ."
Warning: If it is tempting to say that the limit at exists and equals This is incorrect. If the limit does not exist; the notation merely gives information about the way in which the limit fails to exist, i.e. the value of the function "approaches " or increases without bound as .
What can we say about
Separating the limit into and , we obtain
To prove the first statement, for any in the formal definition, we can take and the proof of the second statement is similar.
So the function increases without bound on the right side and decreases without bound on the left side. We cannot say anything else about the two-sided limit or Contrast this with the next example.
What can we say about
Separating the limit into and , we obtain
Since these limits are the same, we have Again, this limit does not, strictly speaking, exist, but the statement is meaningful nevertheless, as it gives information about the behavior of the function near
Then given (A), (B), (C), or (D), equals which of (1), (2), (3), and (4)?
Match the columns:
|(C) if is even, and||(3)|
|(D) if is even, and||(4)|
Note: For example, if (A) correctly matches (1), (B) with (2), (C) with (3), and (D) with (4), then answer as 1234.
Another extension of the limit concept comes from considering the function's behavior as "approaches ," that is, as increases without bound.
The equation means that the values of can be made arbitrarily close to by taking sufficiently large. That is,
There are similar definitions for as well as and so on.
Graphically, corresponds to a vertical asymptote at while corresponds to a horizontal asymptote at
Main Article: Limits by Factoring
Limits by factoring refers to a technique for evaluating limits that requires finding and eliminating common factors.
- Alexandrov, O. Discontinuity. Retrieved September 12, 2005, from https://commons.wikimedia.org/wiki/File:Discontinuity_removable.eps.png