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OBJECTIVES ea SPORTS In football if the, Calculate limits. of polynomial length of a penalty exceeds half, and rational p li c a ti the distance to the offending. functions team s goal line then the ball is moved only. algebraically, half the distance to the goal line Suppose. Evaluate limits 2 5, of functions, one team has the ball at the other team s. 10 yard line The other team in an effort to 1 25, calculator prevent a touchdown repeatedly commits Goal 5 10 15.

penalties After the first penalty the ball, would be moved to the 5 yard line. The results of the subsequent, Penalty 1st 2nd 3rd. penalties are shown in the table, Yard Line 5 2 5 1 25. Assuming the penalties could, continue indefinitely would. the ball ever actually cross the, The ball will never reach the goal line but it will get closer and closer after.

each penalty As you saw in Chapter 12 a number that the terms of a sequence. approach without necessarily reaching it is called a limit In the application. above the limit is the goal line or 0 yard line The idea of a limit also exists for. If there is a number L such that the value of f x gets closer and closer. to L as x gets closer to a number a then L is called the limit of f x as x. Limit of a, approaches a, In symbols L lim f x, Example 1 Consider the graph of the function y f x f x. shown at the right Find each pair of values, a f 2 and lim f x. At the point on the graph where the x coordinate, is 2 the y coordinate is 6 So f 2 6. Look at points on the graph whose O x, x coordinates are close to but not equal to 2. Notice that the closer x is to 2 the closer y is to. 6 So lim f x 6, Lesson 15 1 Limits 941, b f 4 and lim f x.

The hole in the graph indicates that the function does not have a value. when x 4 That is f 4 is undefined, Look at points on the graph whose x coordinates are close to but not equal. to 4 The closer x is to 4 the closer y is to 3 So lim f x 3. You can see from Example 1 that sometimes f a and lim f x are the same. but at other times they are different In Lesson 3 5 you learned about. continuous functions and how to determine whether a function is continuous or. discontinuous for a given value We can use the definition of continuity to make a. statement about limits, Limit of a f x is continuous at a if and only if. Continuous, lim f x f a, Function x a, Examples of continuous functions include polynomials as well as the. functions sin x cos x and a x Also loga x is continuous if x 0. Example 2 Evaluate each limit, a lim x3 5x 2 7x 10. Since f x x 3 5x 2 7x 10 is a polynomial function it is continuous. at every number So the limit as x approaches 3 is the same as the value of. f x at x 3, lim x 3 5x 2 7x 10 33 5 32 7 3 10 Replace x with 3.

27 45 21 10, The limit of x 3 5x 2 7x 10 as x approaches 3 is 7. Since the denominator of is not 0 at x the function is continuous. lim Replace x with, The limit of as x approaches is. Limits can also be used to model real world situations in which values. approach a given value, 942 Chapter 15 Introduction to Calculus. Example 3 PHYSICS According to the special, l Wor theory of relativity developed by. ea Albert Einstein the length of a, moving object as measured by.

p li c a ti an observer at rest shrinks as its, speed increases The difference is. only noticeable if the object is, moving very fast If L0 is the. length of the object when it is at, rest then its length L as measured. by an observer at rest when, information traveling at speed v is given by the. about relativity 2, visit www amc formula L L0 1 2 where c is.

glencoe com the speed of light If the space, shuttle were able to approach the. speed of light what would happen, to its length, We need to find lim L0 1 2. lim L0 1 2 L0 1 2 Replace v with c the speed of light. The closer the speed of the shuttle is to the speed of light the closer the. length of the shuttle as seen by an observer at rest gets to 0. When a function is not continuous at the x value in question it is more. difficult to evaluate the limit Consider the function f x This function is. not continuous at x 3 because the denominator is 0 when x 3 To compute. lim f x apply algebraic methods to decompose the function into a simpler one. x2 9 x 3 x 3, x 3 x 3 Simplify, When computing the limit we are only interested in x values close to 3. What happens when x 3 is irrelevant so we can replace f x with the simpler. expression x 3, lim lim x 3, x 3 x 3 x 3, 3 3 or 6 f x x2 9. The graph of f x indicates that this answer, is correct As x gets closer to 3 the y coordinates.

get closer and closer to but never equal 6 The, limit is 6. Lesson 15 1 Limits 943, Example 4 Evaluate each limit. x 2 2x 8 x 2 x 4, x 4x x x 4, or Replace x with 4, 3 h h2 4h 6. h 0 h h 0 h, lim h2 4h 6, 02 4 0 6 or 6 Replace h with 0. Sometimes algebra is not sufficient to find a limit A calculator may be useful. Consider the problem of finding lim where x is in radians The function is. not continuous at x 0 so the limit cannot be found by replacing x with 0 On. the other hand the function cannot be simplified to help make the limit easier. to find You can use a calculator to compute values of the function for. x values that get closer and closer to 0 from either side that is both less than 0. and greater than 0, Rounded value for table display.

Calculator, Enter the function in the Actual value to 12 decimal places. Y menu and set, Indpnt to Ask in the, TBLSET menu to help The tables below show the expression evaluated for values of x that approach 0. generate these values, sin x sin x, 1 0 841470984808 1 0 841470984808. 0 1 0 998334166468 0 1 0 998334166468, 0 01 0 999983333417 0 01 0 999983333417. 0 001 0 999999833333 0 001 0 999999833333, 0 0001 0 999999998333 0 0001 0 999999998333.

As x gets closer and closer to 0 from either side the value of gets closer. and closer to 1 That is lim 1, 944 Chapter 15 Introduction to Calculus. Example 5 Evaluate each limit, 2 x is in radians, 1 0 45970 1 0 45970. A graphing, calculator or 0 1 0 49958 0 1 0 49958, spreadsheet can 0 01 0 499996 0 01 0 499996. generate more 0 001 0 49999996 0 001 0 49999996, decimal places for. the expression 1 cos x, As x approaches 0 the value of.

2 gets closer to 0 5 so, than shown here x, 0 9 1 0536 1 1 0 95310. 0 99 1 0050 1 01 0 99503, 0 999 1 0005 1 001 0 99950. The closer x is to 1 the closer is to 1 so lim 1, Using a calculator is not a foolproof way of evaluating lim f x You may only. analyze the values of f x for a few values of x near a However the function may. do something unexpected as x gets even closer to a You should use algebraic. methods whenever possible to find limits, GRAPHING CALCULATOR EXPLORATION. You can use a graphing calculator to find a limit WHAT DO YOU THINK. with less work than an ordinary scientific ln x, 3 If you graph y and use TRACE.

calculator To find lim f x first graph the x 1, x a why doesn t the calculator tell you what y is. equation y f x Then use ZOOM and when x 1, TRACE to locate a point on the graph whose. 4 Solve Exercise 2 algebraically Do you get, x coordinate is as close to a as you like The the same answer as you got from the. y coordinate should be close to the value of, graphing calculator. 5 Will the graphing calculator give you the, TRY THESE Evaluate each limit exact answer for every limit problem.

ex 1 x2 4 Explain, 1 lim 2 lim, Lesson 15 1 Limits 945. C HECK FOR U N D E R S TA N D I N G, Communicating Read and study the lesson to answer each question. Mathematics, 1 Define the expression limit of f x as x approaches a in your own words. 2 Describe the difference between f 1 and lim f x and explain when they would. be the same number, 3 MathJournal Write a description of the three methods in this lesson for. computing lim f x Explain when each method would be used and include. examples x a, Guided Practice 4 Use the graph of y f x to find f x.

lim f x and f 0, Evaluate each limit, 5 lim 4x 2 2x 5 6 lim 1 x 2x cos x. x 2 x 2 3x, x 2 x 4 x 0 x 4x, x 2 3x 10 2x 2 5x 2, 9 lim 10 lim. x 3 x 5x 6 x 2 x x 2, 11 Hydraulics The velocity of a molecule of liquid flowing. through a pipe depends on the distance of the molecule. from the center of the pipe The velocity in inches. per second of a molecule is given by the function, v r k R2 r 2 where r is the distance of the molecule. from the center of the pipe in inches R is the radius of. the pipe in inches and k is a constant Suppose for a. particular liquid and a particular pipe that k 0 65. a Graph v r, b Determine the limiting velocity of molecules closer.

and closer to the wall of the pipe, E XERCISES, Practice Use the graph of y f x to find each value. A 12 lim f x and f 2 f x, 13 lim f x and f 0, 14 lim f x and f 3 O x. Evaluate each limit, 15 lim 4x 2 3x 6 16 lim x 3 3x 2 4. B 17 lim 18 lim x cos x, 946 Chapter 15 Introduction to Calculus www amc glencoe com self check quiz. 19 lim 20 lim, x 5 x 5 n 0 n, x 2 3x x 3 3x 2 4x 8.

21 lim 2 2x 22 lim, x 3 x 15 x 1 x 6, h2 4h 4 2x 2 3x. 23 lim 24 lim, h 2 h 2 x 3 x 3 2x 2 x 6, x 3 x22 2x. 25 lim 33 4x 2x, 4x22 26 lim 2, x 0 xx 2x x 0 x x, x 2 2 4 x 1 2 1. C 27 lim 28 lim, x 0 x x 2 x 2, 29 lim 2 30 lim 3, x 2 x 4 x 4 x 64. 31 lim x 32 lim x 4, x 1 x 1 x 4, 33 Find the limit as h approaches 0 of.

34 What value does the function g x approach as x approaches 0. Graphing Use a graphing calculator to find the value of each limit Use radians with. Calculator trigonometric functions, tan 2x ln x, 35 lim 36 lim. x 0 x x 1 ln 2x 1, 1 x 3x sin 3x, 37 lim 38 lim, x 1 x 1 x 0 x 2 sin x. Applications 39 Geometry The area of an ellipse with semi major. and Problem axis a is a a2 c2 where c is the distance from the. Solving foci to the center Find the limit of the area of the. l Wor ellipse as c approaches 0 Explain why the answer a. makes sense, p li c a ti, 40 Biology If a population of tbacteria doubles every 10 hours then its initial. hourly growth rate is lim t where t is the time in hours Use a. calculator to approximate the value of this limit to the nearest hundredth. Write your answer as a percent, 41 Critical Thinking. Does lim sin exist That is can you say, lim sin L for some real number L Explain why or why not.

42 Critical Thinking You saw in Example 5 that lim. 2 0 5 That is for, 1 cos x x2, values of x close to 0. 2 0 5 Solving for cos x we get cos x 1, a Copy and complete the table by using a calculator Round to six decimal. places if necessary, x 1 0 5 0 1 0 01 0 001, b Is it correct to say that for values of x close to 0 the expression 1 is a. good approximation for cos x Explain, Lesson 15 1 Limits 947. 43 Physics When an object such as a bowling ball is dropped near Earth s. surface the distance d t in feet that the object falls in t seconds is given. by d t 16t 2 Its velocity in feet per second after 2 seconds is given by. lim Evaluate this limit algebraically to find the velocity of the bowling. ball after 2 seconds You will learn more about the relationship between distance. and velocity in Lesson 15 2, 44 Critical Thinking Yoshi decided that lim 1 x x is 0 because as x approaches.

0 the base of the exponential expression approaches 1 and 1 to1 any power is 1. a Use a calculator to help deduce the exact value of lim 1 x x. b Explain where Yoshi s reasoning was wrong, Mixed Review 45 Botany A random sample of fifty. acorns from an oak tree in the park, reveals a mean diameter of 16 2. millimeters and a standard deviation of, 1 4 millimeters Find the range about. the sample mean that gives a 99, chance that the true mean lies within it. Lesson 14 5, 46 Tess is running a carnival game that involves spinning a wheel The wheel has.

the numbers 1 to 10 on it What is the probability of 7 never coming up in five. spins of the wheel Lesson 13 6, 47 Find the third term of x 3y 5 Lesson 12 6. 48 Simplify 16y 8 4 Lesson 11 1, 49 Write the equation of the ellipse if the endpoints of the major axis are at 1 2 and. 9 2 and the endpoints of the minor axis are at 5 1 and 5 5 Lesson 10 3. 50 Graph the polar equation r 3 Lesson 9 1, 51 Write the ordered pair that represents. WX for W 4 0 and X 3 6 Then find, the magnitude of. WX Lesson 8 2, 52 Transportation A car is being driven at 65 miles per hour The car s tires.

have a diameter of 25 inches What is the angular velocity of the wheels in. revolutions per second Lesson 6 2, 53 Use the unit circle to find the value of csc 270 Lesson 5 3. 54 Determine the rational roots of the equation 12x 4 11x 3 54x 2 18x 8 0. Lesson 4 4, 55 Without graphing describe the end behavior of the function y 4x 5 2x 2 4. Lesson 3 5, 56 Find the value of the determinant, Lesson 2 5. 57 Geometry Determine whether the figure with vertices at 0 3 8 4 2 5. and 10 4 is a parallelogram Explain Lesson 1 5, 58 SAT Practice Grid In If 2n 8 what is the value of 3n 2. 948 Chapter 15 Introduction to Calculus Extra Practice See p A55. GRAPHING CALCULATOR EXPLORATION, 15 2A The Slope of a Curve.

A Preview of Lesson 15 2, OBJECTIVE Recall from Chapter 1 that the slope of a line is a measure of its steepness. Calculus Calculus is one of the most important areas of mathematics There are two branches of calculus differential calculus and integral calculus Differential calculus deals mainly with variable or changing quantities Integral calculus deals mainly with finding sums of infinitesimally small quantities This generally involves finding a limit Chapter 15 the only chapter in Unit 5