At the end of this lesson, you, as a successful student, will be able to understand and to use in activities
At the end of this session you will be able to
Calculus is built on the concept of the limit. Some mathematicians say that calculus is the study of limits. Therefore, right now we need to learn, at least informally, what a limit is.
COMPARING EVALUATION OF FUNCTION TO EVALUATING A LIMIT
When you evaluate a function, you are finding the dependent variable value at a particular value of the independent variable.
For example, if f(x) = 3x - 2 and you want to know f(3), then you substitute x = 3 into the equation. That evaluation of f(3) gives us a value at x = 3. We say, "We evaluate the function at x = 3." Therefore, f(3) is 3 x 3 -2 or f(3) = 7, meaning that we have the point (3,f(s)) = (3, 7).
Thus, when we want to evaluate a function, we want to know the value of f(x) when a value of x is given.
When you want to evaluate a limit, you are determining the dependent variable value when the values of the independent variable get very, very, very close to a value. When you want to evaluate a limit, we say that we want the value of f(x) when "x gets close to a value" or "x approaches a value", not when x has the value.
For example, if we want the limit of (x - 1)/(x2 - 1) as "x approaches 1", we want to know what is the fraction when x takes on values closer and closer to 1, but never becomes 1. When evaluating the limit, we are not substituting in the value x gets close to, but rather values of x very close to 1. In this example, the limit is .5 as x approaches 1.
Notice that f(1) is undefined when f(x) = (x - 1)/(x2 - 1) because f(1) = 0/0. This we see that the limit can exist when the function does not have a defined value!
INFORMAL DEFINITION OF A LIMIT
WHAT IS THE LIMIT?
A limit is a number. Always, the LIMIT is a NUMBER! The LIMIT is a number f(x) approaches or a number that f(x) actually is when x approaches a value. Remember the independent variable x does not attain the value, but only approaches the value. In other words, the independent variable x assumes values sufficiently close to a value.
Always the LIMIT is a NUMBER for the dependent variable, if the limit exists, when the independent variable approaches a value.
You can now go to the Session 4 assignments to continue your assignment.
At the end of this session you will be able to
EVALUATION WITH THE GRAPHING CALCULATOR
TI-83 Plus and TI-84 Plus Calculator:
This is the process for completing successive evaluations of a function using your graphing calculator.
- Enter the function in y =.
- Insure that the value of x, where we desire the limit, is in the window domain.
- Select 2nd CALC.
- Select 1:VALUE or just press the ENTER key.
- Now enter the value for the independent variable.
- Continue entering values until you have completed all of your evaluations.
TI-86 Calculator:
This is the process for completing successive evaluations of a function using your graphing calculator.
- Enter the function in y =.
- Insure that the value of x, where we desire the limit, is in the window domain.
- From the y(x) = menu, select MORE MORE.
- Select F1 EVAL.
- Now enter the value for the independent variable.
- Continue entering values until you have completed all of your evaluations.
LIMIT FROM TABLES
Using the evaluation functions of your graphing calculator, you can create easily a table of values. Continuing entering values in the calculator for the independent variable, closer and closer to the desired value where we desire the limit, until you are sure that you can determine the limit.
In process to find the limit with f(x) = (x - 1)/(x2 - 1) as "x approaches 1", I made the following tables using the evaluation key of the graphing calculator.
x < 1 f(x) x > 1 f(x) 0.5 0.666667 1.5 0.400000 0.9 0.526316 1.1 0.476190 0.99 0.502513 1.01 0.497512 0.999 0.500250 1.001 0.499750 0.9999 0.500025 1.0001 0.499975 Using your calculator, evaluate the function at these values for x from the table above. Do you obtain the same values of f(x)? What does it appear that the limit L is?
LIMIT FROM A GRAPH
How can we guess the limit from a graph?
Answer: If we want the limit of f(x) as x approaches a, then graph f(x) with x = a in the domain as you define the "window" in your graphing calculator.
- Enter the rule for the function into the calculator.
- Using the trace function, get as close as you can to x = a.
- Then, use the zoom function to zoom in.
- Repeat trace and zoom in until you are relatively sure that you know the limit.
In the tracing process, go to values on both sides of the place where we want to find the limit.
Now, using your graphing calculator, try this process to find the limit with f(x) = (x - 1)/(x2 - 1) as "x approaches 1".
You should obtain the limit L = .5. Did you?
You can now go to the Session 4 assignments to continue your assignment.
At the end of this session you will be able to
PROPERTIES (THEOREM) OF LIMITS
Theorem 1 on page 84 of your textbook states the properties of limits. The description of what each of these rules state is given immediately below each rule. You should be able to describe each one of these six rules without looking at your textbook.
DETERMINATE LIMIT
In your textbook, when you read on page 86 Theorems 2 and 3 and practiced Example 2, you might think that evaluating a limit is the same as evaluating a function. That is not exactly true.
To see what the theorems are saying, look at Example 2 on page 86. That symbolism is saying that as x approaches -1, then x cubed approaches (-1) cubed and x squared approaches (-1) squared. Using the limit rules we, in this situation, obtain the same result as you would if you evaluated the function at x = -1, leading you to incorrectly believe that evaluating a limit by using substitution is exactly the same as evaluating a function.
When we can use Theorems 2 and 3 from page 86 we have a determinate limit. Notice that the theorems state limits "can be" found by substitution, which implies there are times you cannot find the limit by substitution. The next discussion will illustrate when you cannot use the theorems.
INDETERMINATE LIMIT
In arithmetic, including when you are evaluation a function, you know that division by zero is not possible. When evaluating limits, almost never will you encounter that difficulty because your independent value is "approaching" a value. Sometimes your limit will be 0/0, making you suspect that you have no answer, but most of the time you will have a limit. We have a name for this situation.
An "indeterminate limit" is one in which you intuitively obtain 0 / 0 when you evaluate a limit. Frequently students confuse this with the impossibility of dividing by zero in arithmetic. However, in a limit we are evaluating with values "sufficiently close" to an independent variable value and we are not dividing by zero.
"Indeterminate limit 0/0" means that we have not yet determined the limit and must perform a mathematical process to "evaluate the limit".
Examples 3 and 4 on pages 86 and 87 of your textbook illustrate indeterminate limits.
ALGEBRAICALLY EVALUATING INDETERMINATE LIMITS
When you evaluate a function, you can never have a zero in the denominator because division by zero never gives you a result. In other words, in arithmetic you can never divide by zero and, therefore, when evaluating a function, you can never obtain a zero in the denominator.
However, when you intuitively obtain 0/0 while evaluating limits, recall that you are evaluation the function when the values of the independent variable are sufficiently close to a given value, not at the value.
In calculus, we will learn three major methods of evaluating indeterminate limits, but only two methods in this unit. The methods we will learn now are factoring and rationalizing.
In factoring, we factor the numerator and denominator and divide common factors from the numerator and denominator. Most of the time we can then evaluate the limit of the resulting function. Example 3 on page 86 illustrates the factoring technique.
In rationalizing, we rationalize either the numerator or denominator, where we find a radical. This process usually creates common factors that we can divide out of the numerator and denominator. Example 4 on page 87 illustrates the rationalizing technique.
You can now go to the Session 4 assignments to continue your assignment.
At the end of this session you will be able to
f(x) APPROACHES L AS x APPROACHES a+
This situation is called a right-hand limit and is defined at the middle of page 102 of your textbook. The figure 2.22 on page 102 explain intuitively the concept. Notice that you are sufficiently close to x = a with values of x greater than a. Example 2 on page 103 illustrates well right-hand limits.
f(x) APPROACHES L AS x APPROACHES a-
This situation is called a left-hand limit and is defined at the middle of page 102. The figure 2.22 on page 102 explain intuitively the concept. Notice that you are sufficiently close to x = a with values of x less than a. Example 2 on page 103 illustrates well left-hand limits.
EVALUATE ONE-SIDED LIMIT
You evaluate one-sided limits exactly as you do other limits except you can only use values sufficiently close on one side of the independent value in question. Again, Examples 1, 2, and 4 illustrate evaluating one-sided limits on pages 103 and 104. Make sure you know how the textbook obtained all of the answers.
EXISTENCE OF A LIMIT
To have a limit existing, both the right-hand limit and left-hand limit must exist. Also, to have a limit existing, both the right-hand limit and left-hand limit must be the same. That is what Theorem 6 on page 103 of your textbook is stating.
Example 2 on page 103 illustrates all of the different situations that can arise for the existence and non existence of a limit.
You can now go to the Session 4 assignments to continue your assignment.
At the end of this session you will be able to
LIMIT OF ((sin t)/t) WHERE t APPROACHES 0
As you study pages 105 and 106 of your text book, learning that the limit of ((sin t)/t) where t approaches 0 is 1 is much more important that learning how to derive the result using the Sandwich Theorem. Just knowing that the Sandwich Theorem is used in the proof is sufficient knowledge of the proof.
Knowing that the limit is 1 is very important as you study calculus.
EVALUATE LIMITS USING LIMIT OF ((sin t)/t) WHERE t APPROACHES 0
To evaluate limits using the limit of ((sin t)/t) where t approaches 0 is 1 requires you to change any function into the form (sin t)/t. For example, when you look at Example 6(b) on page 107 of your textbook, you see that you have (sin(2x))/(2x) which is in the form ((sin t)/t).
For example, suppose we want the limit of (sin (3x))/x as x approaches 0.
What would we need to multiply the numerator and denominator by to obtain the form ((sin t)/t)?
Then you will have the limit of (3 sin (3x))/(3x) as x approaches 0. When x approaches 0, you also have 3x approaching zero and our function is in the form of ((sin t)/t) where 3x = t.
Using the properties of limits, you have 3 limit of (sin (3x))/(3x) as 3x approaches 0. What is your final answer?
For a second example, try finding the limit of (sin (3x))/(4x) as x approaches 0.
What would you need to multiply the numerator and denominator by to obtain the form ((sin t)/t)?
Then what do you have?
Now what do we need to factor from the numerator and what from the denominator to have the form ((sin t)/t)?
Finally, what is the limit?
You can now go to the Session 4 assignments to continue your assignment.