The Limit of a Function
Site: | Saylor Academy |
Course: | MA005: Calculus I |
Book: | The Limit of a Function |
Printed by: | Guest user |
Date: | Tuesday, October 22, 2024, 4:22 AM |
Description
Read this section for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1-4.
The Idea, Informally
Calculus has been called the study of continuous change, and the limit is the basic concept which allows us to describe and analyze such change. An understanding of limits is necessary to understand derivatives, integrals, and other fundamental topics of calculus.
The limit of a function describes the behavior of the function when the variable is near, but does not equal, a specified number (Fig. 1). If the values of get closer and closer, as close as we want, to one number as we take values of very close to (but not equal to) a number , then we
say "the limit of , as x approaches , is " and we
write " (The symbol " " means "approaches" or "gets very close to").
is a single number that describes the behavior (value) of f AT the point .
is a single number that describes the behavior of NEAR, BUT NOT AT, the point .
If we have a graph of the function near , then it is usually easy to determine .
Example 1: Use the graph of in Fig. 2 to determine the following limits:
Solution: (a) . When is very close to 1 , the values of are very close to In this example, it happens that , but that is irrelevant for the limit. The only thing that matters is what happens for close to 1 but .
(b) is undefined, but we only care about the behavior of for close to 2 and not equal to 2. When is close to 2 , the values of are close to 3 . If we restrict close enough to 2 , the values of will be as close to 3 as we want, so .
(c) When is close to 3 (or as approaches the value 3 ), the values of are close to 1 (or approach the value 1), so . For this limit it is completely irrelevant that , We only care about what happens to for close to and not equal to 3.
(d) This one is harder and we need to be careful. When is close to 4 and slightly less than 4 ( is just to the left of 4 on the -axis), then the values of are close to But if is close to 4 and slightly larger than 4 then the values of are close to If we only know that is very close to 4, then we cannot say whether will be close to 2 or close to 3 - it depends on whether is on the right or the left side of In this situation, the values are not close to a single number so we say does not exist. It is irrelevant that The limit, as approaches 4, would still be undefined if was 3 or 2 or anything else.
Practice 1: Use the graph of in Fig. 3 to determine the following limits:
Example 2: Determine the value of .
Solution: We need to investigate the values of when is close to 3. If the values get arbitrarily close to or even equal some number , then will be the limit. One way to keep track
of both the and the values is to set up a table and to pick several values which are closer and closer (but not equal) to We can pick some values of which approach 3 from the left, say ,
and , and some values of which approach 3 from the right, say , and . The only thing important about these particular values for is that they get closer and closer to 3 without equaling
You should try some other values "close to 3" to see what happens.
Instead of using a table of values, we could have graphed for close to 3, Fig. 4, and used the graph to answer the limit question. This graphic approach is easier, particularly if you have a calculator or computer do the graphing work for you, but it is really very similar to the "table of values" method: in each case you need to evaluate at many values of near 3.
You might have noticed that if we just evaluate , then we get the correct answer That works for this particular problem, but it often fails. The next example illustrates the difficulty.
Example 3: Find . (Same as Example 2 but with ).
Solution: You might try to evaluate at , but is not defined at . It is tempting, but wrong, to conclude that this function does not have a limit as approaches 1.
Table Method: Trying some "test" values for which get closer and closer to 1 from both the left and the right, we get
The function is not defined at , but when is close to 1, the values of are getting very close to 3. We can get as close to 3 as we want by taking very close to 1 so
Graph Method: We can graph for close to 1 , Fig. 5, and notice that whenever is close to 1 , the values of are close to is not
defined at , so the graph has a hole above , but we only care about what is doing for close to but not equal to 1.
Algebra Method: We could have found the same result by noting that as long as . (If , then so it is valid to divide
the numerator and denominator by the factor ) The "" part of the limit means that is close to 1 but not equal to , so our division step is valid and
Source: Dave Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2011/11/2-2FunctionLimit.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.
Three Methods for Evaluating Limits
The Algebra Method
The algebra method involves algebraically simplifying the function before trying to evaluate its limit.
Often, this simplification just means factoring and dividing, but sometimes more complicated algebraic or even trigonometric steps are needed.
The Table and Graph Methods
To evaluate a limit of a function as approaches , the table method involves calculating the values of for "enough" values of very close to so that we can "confidently" determine which value is approaching. If is well-behaved, we may not need to use very many values for . However, this method is usually used with complicated functions, and then we need to evaluate for lots of values of .
A computer or calculator can often make the function evaluations easier, but their calculations are subject to "round off" errors. The result of any computer calculation which involves both large and small numbers should be viewed with some suspicion. For example, the function
for every value of , and my calculator gives the correct answer for some values of , and and both equal 1.
But my calculator says so it evaluates to be 0, definitely an incorrect value. Your calculator may evaluate correctly, but try or .
Calculators are too handy to be ignored, but they are too prone to these types of errors to be believed uncritically. Be careful.
The graph method is closely related to the table method, but we create a graph of the function instead of a table of values, and then we use the graph to determine which value is approaching.
Which Method Should You Use?
In general, the algebraic method is preferred because it is precise and does not depend on which values of x we chose or the accuracy of our graph or precision of our calculator. If you can evaluate a limit algebraically, you should do so. Sometimes, however, it will be very difficult to evaluate a limit algebraically, and the table or graph methods offer worthwhile alternatives. Even when you can algebraically evaluate the limit of a function, it is still a good idea to graph the function or evaluate it at a few points just to verify your algebraic answer.
The table and graph methods have the same advantages and disadvantages. Both can be used on very complicated functions which are difficult to handle algebraically or whose algebraic properties you don't know.
Often both methods can be easily programmed on a calculator or computer. However, these two methods are very time-consuming by hand and are prone to round off errors on computers. You need to know how to use these methods when you can't figure out how to use the algebraic method, but you need to use these two methods warily.
Solution: The function in each limit is the same but is approaching a different number in each of them.
(a) Since , we know that is getting closer and closer to 0 so the values of the and terms get as close to 0 as we want. The numerator approaches 6 and the denominator approaches 2, so the values of the whole function get arbitrarily close to , the limit.
(b) As approaches , the numerator and denominator approach 0, and a small number divided by a small number can be almost anything - the ratio depends on the size of the top compared to the bottom. More investigation is needed.
Table Method: If we pick some values of close to (but not equal to) , we get the table
|
|
|
|
↓ | ↓ | ↓ | ↓ |
-2 | 0 | 0 | -1 |
Even though the numerator and denominator are each getting closer and closer to , their ratio is getting arbitrarily close to which is the limit.
Graph Method: The graph of in Fig. 6 shows that the values of are very close to when the -values are close to .
We know so , and we can divide the top and bottom by . Then so as .
One-Sided Limits
Sometimes, what happens to us at a place depends on the direction we use to approach that place. If we approach Niagara Falls from the upstream side, then we will be 182 feet higher and have different worries than if we approach from the downstream side. Similarly, the values of a function near a point may depend on the direction we use to approach that point. If we let approach 3 from the left ( is close to and , then the values of equal (Fig. 7). If we let approach 3 from the right ( is close to 3 and ), then the values of equal .
On the number line we can approach a point from the left or right, and that leads to one-sided limits.
Definition of Left and Right Limits:
Example 5: Evaluate and .
Solution: The left-limit notation requires that be close to 2 and that be to the left of 2, so .
If is close to 2 and is to the right of 2, then so and .
The graph of is shown in Fig. 8.
If the left and right limits have the same value, , then the value of is close to whenever is close to , and it does not matter if is left or right of so . Similarly, if , then is close to whenever is close to and less than and whenever is close to and greater than , so . We can combine these two statements into a single theorem.
One-sided limits are particularly useful for describing the behavior of functions which have steps or jumps.
To determine the limit of a function involving the greatest integer or absolute value or a multiline definition, definitely consider both the left and right limits.
Practice 3: Use the graph in Fig. 9 to evaluate the one and two-sided limits of at , and .
Find the one and two-sided limits of at and .