10.2 Basic Concepts of Limits

Transcription

1 10.2 Basic Concepts of Limits Question 1: How do you evaluate a limit from a table? Question 2: How do you evaluate a limit from a graph? In this chapter, we ll examine the concept of a limit. in its simplest interpretation a limit indicates how the output of a function behaves as the input is changed. Limits help us to spot patterns in this behavior. We ll begin this journey by using tables and graphs to relate the input and output of a function. These forms make it easy to spot patterns in the relationship. Unfortunately, these representations take a bit of work to set up. In later sections we ll learn other methods that can help us to evaluate a limit. 1

2 Question 1: How do you evaluate a limit from a table? Suppose we have a rational function f( x) 2 x 9 x 3. We can examine the behavior of this function by constructing a table of values. x f (x) undefined This function is undefined at x 3 so we cannot find a function value there. However, we can find function values near x 3.This table shows how the functions values change for different values of x near x 3. In this table, x values on the left of 3 are in red. Values on the right side of 3 are in blue. Let s take a closer look at x values on the left side of x 3. As x gets closer to 3 from the left x f (x) undefined f (x) gets closer to 6 On the left side of 3, the function values get closer and closer to 6 as x gets closer to 3. Mathematically, we say that the limit as x approaches 3 from the left is 6. A limit represents the relationship between x and f (x) as x approaches some value. This relationship is represented symbolically using the symbol lim. 2

3 lim f ( x) 6 x3 As x approaches 3 from the left on f(x), the function values approach 6 We read this as, the limit of f (x) as x approaches 3 from the left is 6. The value of the limit is determined by x values that are closer and closer to 3, not the value at x 3. We will examine situations later in this section when we will use the value at a point to find the limit. In general, the value of the limit as x approaches some value is not equal to the value of the function at that x value. In the case of even defined at x 3 f( x) 2 x 9 x 3, the function is not A function f (x) has a limit L as x approaches a from the left, if the value of f (x) can be made arbitrarily close to L as x is made arbitrarily close to a from values on the left of a (but not equal to a). We symbolize this relationship by writing lim f ( x) L xa We can also evaluate a limit on the right side of x 3 on the function f( x) 2 x 9 x 3. As x gets closer to 3 from the right x f (x) undefined f (x) gets closer to 6. 3

4 Based on this table, we see that as x gets closer and closer to 3 on the right, f (x) gets closer and closer to 6. This means that 6 x3 The plus sign behind the x value indicates that we are approaching the value from the right side of 3. We read this as the limit of f (x) as x approaches 3 from the right is 6. A function f (x) has a limit L as x approaches a from the right, if the value of f (x) can be made arbitrarily close to L as x is made arbitrarily close to a from values on the right of a (but not equal to a). We symbolize this relationship by writing lim f ( x) L xa Example 1 Find a Limit Using a Table Let f (x) be the piecewise function 3 2 x if 0 x100 f( x) 0.9x60 if x100 Use a table to find each of the limits below. a. lim f ( x) x100 Solution In this piecewise function, two expressions define the function. For x values in the interval [0,100), the expression 3 2 x gives the y values for the function. We can use this expression to construct a table for values of x slightly to the left of x

5 x f (x) As the x values in the table get closer and closer to 100, the corresponding f (x) values get closer and closer to 150. Based on this observation, we conclude that the left hand limit is 150 x100 b. lim f ( x) x100 Solution For x values in the interval [100, ), the expression 0.9x 60 gives the y values for the function. This expression can help us to form the following table of values slightly to the right of x 100. x f (x) As x values get closer and closer to 100, the values for f (x) get closer and closer to 150. The right hand limit is 150 x100 c. lim f ( x) x100 Solution Since the left hand and right hand limits are both equal to 150, the two-sided limit is also equal to 150, 150 x100 5

6 Examples like the ones we have examined might lead you to believe that all limits are equal to some value. However, this is not always the case. In the next two examples, the limits do not exist. Example 2 Find a Limit Using a Table Let g(x) be the piecewise function x500 if 0 x1000 gx ( ) 0.2x1350 if x1000 Use a table to find each of the limits below. a. lim gx ( ) x1000 Solution In this limit, x is approaching 1000 from values on the left. On this side of 1000, the function is defined by the expression x 500. Let s examine the behavior of this expression slightly to the left of x g(x) As x approaches 1000, the values of gx ( ) get closer and closer to This indicates that the left hand limit is lim gx ( ) 1500 x1000 b. lim gx ( ) x1000 Solution On the right hand side of 1000, the function is defined by the expression 0.2x The table below shows values in that region. 6

7 x g (x) Based on this table, as x approaches 1000 from the right, the function approaches Symbolically, we write lim gx ( ) 1550 x1000 c. lim gx ( ) x1000 Solution In the previous parts, we evaluated the limits from the left and right. Since the limit from the left is not equal to the limit from the left, the two-sided limit does not exist. Example 3 Find a Limit Using a Table Use a table to evaluate the limit lim x3 x x 3 Solution Since this is a left hand limit, start by constructing a table of values slightly to the left of x 3. x x x ,999 As x approaches 3, the value of the function gets more and more negative. Because the function is not approaching a value, we say that the limit does not exist. 7

8 Since it does the by growing large and negative, we write lim x3 x x 3 For limits that do not exist and do it by becoming more and more positive, we write that the limit is equal to. The infinity symbol always indicates that the limit does not exist because the values are becoming larger and larger. 8

9 Question 2: How do you evaluate a limit from a graph? In the question before this one, we used a table to observe the output values from a function as the input values approach some value from the left of right. With a little practice, we can evaluate limits using a graph to find the values of a function. Suppose we have the graph of a function like the one below. We can use this graph to evaluate the two-sided limit. x2 As with the limits we calculated from tables, we must evaluate the one-sided limits near x 2. To calculate the limit, x2 we must examine the graph at x values that are slightly smaller than x 2. 9

10 Figure 1 - As the x values get closer and closer to 2 from values slightly smaller than 2, the y values approach 4. In Figure 1, a red dashed vertical line is positioned slightly to the left of 2. The height of the line indicates the y value at that x value. A red dashed horizontal line locates the y value on the graph. As the vertical line moves closer and closer to 2, the horizontal line gets closer and closer to the y value 4. This means the limit as x approaches 2 from the left is 4 or 4. x2 The same strategy allows us to solve the one-sided limit x2 10

11 Figure 2 - As the x values get closer and closer to 2 from values slightly larger than 2, the y values approach 4. The red dashed vertical line in Figure 2 locates an x value slightly larger than 2. The red dashed horizontal line gives the corresponding value on the y axis. As the vertical line moves closer and closer to 2, the horizontal line moves closer and closer to 4. In other words, for x values closer and closer to 2, the y values are closer and closer to 4. The limit from the right is 4. x2 Since the limits from the left and right are both equal to 4, the two-sided limit is also equal to 4, 4. x2 11

12 Example 4 Find the Limit Graphically Suppose f( x) is given by the graph below. Evaluate each of the limits below. a. x1 Solution To evaluate this limit, we need to examine y values on the graph as x gets closer and closer to 1 from the left side of 1. This region of the graph is shown in the graph to the below. 12

13 Let us locate an x value and its corresponding y value in this region. y x Notice that as x moves horizontally closer and closer to 1, the corresponding y value moves vertically closer and closer to 1. This tells us that 1. Notice that the y value at x 1, f (1) 2, is not the x1 same as the limit. 13

14 b. x1 Solution In this one sided limit, the x values are on the right side of 1. y x As the point moves to the left towards x 1, the point moves up vertically towards 1. This means that the closer the point gets to x 1, the closer the y value gets to 1 or 1 x1 c. x1 Solution For the two sided limit to exist, the one sided limits must be equal. In this case they are both equal to 1. Since they are both equal to 1, the two sided limit is also equal to 1, lim f ( x) 1 x1 Notice that none of these limits have anything to do with the fact that f (1) 2. This is because we are using x values approaching 1, not equal to 1. 14

15 Example 5 Find the Limit Graphically Suppose f( x) is given by the graph below. Evaluate each of the limits below. a. x1 Solution To left of x 1, the graph looks like the graph in Example 1. 15

16 y x Notice that as x moves horizontally closer and closer to 1, the corresponding y value moves vertically closer and closer to 1. This tells us that 1. x1 b. x1 Solution As the point moves to the left towards x 1, the point moves up vertically towards 2. 16

17 y x This means that the closer the point gets to x 1, the closer the y value gets to 2 or 2. x1 c. x1 Solution For the two sided limit to exist, the one sided limits must be equal. In this case, they are not equal. From the left side the limit is equal to 1 and from the right side the limit is equal to 2, so does not exist x1 The vertical gap in the graph at x 1is what leads to different values in the one sided limits. In Example 1 there was a horizontal gap at x 1, but not a vertical gap since the two pieces of the graph come together at x 1. In each of these examples, we evaluate the one-sided limits to find the two-sided limit. If the one-sided limits are equal to some value, the two-sided limit is equal to the same 17

18 value. If the one-sided limits do not match, the two-sided limit does not exist. In the next example, we examine a function for which the one-sided limit does not exist. Example 6 Find the Limit Graphically Suppose f( x ) is given by the graph below. Evaluate the limit. x5 Solution This function has a vertical asymptote at x 5. The vertical asymptote is shown on the graph as a blue dashed line. The one-sided limit is a left hand limit. Locate points on the left side of x 5 with red dashed lines. 18

19 As the vertical line gets closer and closer to 5, the horizontal line gets higher and higher. This indicates that the y values do not get closer to any value as x gets closer to 5 from the left. The one-sided limit does not exist. 19