Math 1314 Lesson 4 Limits

Transcription

1 Math 1314 Lesson 4 Limits What is calculus? Calculus is the study of change, particularly, how things change over time. It gives us a framework for measuring change using some fairly simple models. In this course, we will address two basic questions in this course. 1. How can we find the slope of a line that is tangent to an arbitrary curve at a given point?. How can we find the area of a region such as the one pictured below? Both of these areas of inquiry rely on the concept of a it. We will begin our study of calculus by looking at its. Lesson 4 Limits 1

2 Limits Finding a it amounts to answering the following question: What is happening to the y-value of a function as the -value approaches a specific target number? If the y-value is approaching a specific number, then we can state the it of the function as gets close to the target number. Notation: is short for it. If we write (y value) as the values approach. it means, the it of f() The it formally defined: We say that a function f has it L as approaches the target number a, written L if the value f() can be made as close to the number L as we please by taking sufficiently close to (but not equal to) a. Note that L is a single real number. Otherwise, the it fails to eist (DNE = Does Not Eist). Eample 1: Given the graph of f() below. Find: a 0 b. c. Lesson 4 Limits

3 Eample : Given the following function, what is the it of the function as approaches 1? f( ) 5 3, if 1 3 9, if 1 Eample 3: Let f 3, if 0 10, if 5 ( ), if 0 <5, find. 5 As in Eample 1 part b, we have seen a it fails to eist when it approaches two different y-values numbers from each side of a target number (the given value). A it also fails to eist when at the target number for the -value, the graph of the function has an asymptote. The y-values will approach. Lesson 4 Limits 3

4 Evaluating Limits Most often, we will not have a graph given, and will need to find a it by algebraic methods. Properties of its: Suppose 1. L and g( ) M. Then, r r r [ f ( )] [] L for any real number r. a a. cf ( ) c cl for any real number c. f g f g L M 4. [ f ( ) g( )] [][ g( )] LM. f ( ) L 5., provided M 0. g( ) g( ) M 3. Substitution Eample 4: Evaluate: 3 1 Eample 5: Evaluate 6: Evaluate: 55 1 Eample 7: Evaluate: Eample 8: Evaluate: When substitution gives you a value in the form 0 k, where k is any non-zero real number the it DNE. Lesson 4 Limits 4

5 Indeterminate Forms 0 What do you do when substitution gives you the value? 0 This is called an indeterminate form. It means that we are not done with the problem! We must try another method for evaluating the it!! Once we determine that a problem is in indeterminate form, we can use GGB to find the it. Command: it[<function>,<value>] This command can also be used when evaluating more complicated its. Eample 9: The following eamples are of the indeterminate form. Use GGB to evaluate the it. Begin by entering the function into GGB. a Command: Answer: b. 4 4 Command: Answer: Lesson 4 Limits 5

6 So far we have looked at problems where the target number is a specific real number. Sometimes we are interested in finding out what happens to our function as increases (or decreases) without bound. Limits at Infinity Consider the function f ). As the value of get larger and larger, f() 1 ( approaches. We can see this by looking at the table below or its graph. f() ,000 1,000, If a it at infinity eists and it s equal to a single real number L then they are written as L or L. Limits at infinity problems often involve rational epressions (fractions). The technique we can use to evaluate its at infinity is to recall some rules from Algebra used to find horizontal asymptotes. These rules came from its at infinity so they ll surely work for us here. The highest power of the variable in a polynomial is called the degree of the polynomial. We can compare the degree of the numerator with the degree of the denominator and come up with some generalizations. If the degree of the numerator is smaller than the degree of the denominator, then f ( ) 0. g( ) If the degree of the numerator is the same as the degree of the denominator, then you can f ( ) find by making a fraction from the leading coefficients of the numerator and g ( ) denominator and then reducing to lowest terms. If the degree of the numerator is larger than the degree of the denominator, then it s best to use the it command in GGB. Lesson 4 Limits 6

7 Eample 10: Evaluate: Eample 11: Evaluate: 99 Eample 1: Evaluate: Command: Answer: Eample 13: 3 39 Command: Answer: Lesson 4 Limits 7

8 Applications Involving Limits Eample 14: The average cost in dollars of constructing each skateboard when skateboards are 13,500 produced can be modeled by the function C ( ) 1.5. What is the average cost per skateboard if the number of boards produced gets larger? Eample 15: The population of deer in a forest is epected to grow according to the model 17t Pt (), where t is given in years since the population was first counted, and P(t) is t t 7 given in hundreds. As more time passes, what should be the deer population in the forest? Lesson 4 Limits 8