Lesson A Limits. Lesson Objectives. Fast Five 9/2/08. Calculus - Mr Santowski

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1 Lesson A Limits Calculus - Mr Santowski 9/2/08 Mr. Santowski - Calculus 1 Lesson Objectives 1. Define its 2. Use algebraic, graphic and numeric (AGN) methods to determine if a it exists 3. Use algebraic, graphic and numeric methods to determine the value of a it, if it exists 4. Use algebraic, graphic and numeric methods to determine the value of a it at infinity, if it exists 5. Be able to state and then work with the various laws of its 6. Apply its to application/real world problems 9/2/08 Mr. Santowski - Calculus 2 Fast Five You will get a hand out of the following three slides. You and your table have 5 minutes to answer the it questions and then defend your answers (if challenged) 9/2/08 Mr. Santowski - Calculus 3 1

2 Fast Five - Limits and Graphs 9/2/08 Mr. Santowski - Calculus 4 Fast Five - Limits and Graphs (A) Find function values at the following: (i) f(-10) = (ii) f(-4) = (iii) f(2) = (iv) f(7) = 9/2/08 Mr. Santowski - Calculus 5 Fast Five - Limits and Graphs Find the it of the function f(x) at the following values: (i) the it of f(x) at x = -10 is (ii) the it of f(x) at x = -6 is (iii) the it of f(x) at x = -4 is (iv) the it of f(x) at x = 0 is (v) the it of f(x) at x = 2 is (vi) the it of f(x) at x = 4 is (vii) the it of f(x) at x = 6 is (i) the it of f(x) at x = 7 is (i) the it of f(x) at x = 10 is 9/2/08 Mr. Santowski - Calculus 6 2

3 Fast Five - Limits (Lesson #2) Consider the following it(s): f (x) if x f (x) = 1 x n f (x) if x 0 f (x) = 1 x n Determine the iting value(s) if possible. 9/2/08 Mr. Santowski - Calculus 7 (A) Introduction to Limits Let f be a function and let a and L be real numbers. If 1. As x takes on values closer and closer (but not equal) to a on both sides of a, the corresponding values of f(x) get closer and closer (and perhaps equal) to L; and 2. The value of f(x) can be made as close to L as desired by taking values of x close enough to a; Then L is the LIMIT of f(x) as x approaches a Written as x a f(x) = L 9/2/08 Mr. Santowski - Calculus 8 (A) Introduction to Limits We will work with g(x) = x 3 8 x 2 and consider the function behaviour at x = 2 We can express this idea of function behaviour at a point using it notation as x 3 8 x 2 x 2 9/2/08 Mr. Santowski - Calculus 9 3

4 (A) Introduction to Limits We will explore the it in a variety of ways: first using a ToV So notice what happens to the function values as x gets closer to 2 from both sides (RS 2.01, 2.02 & LS 1.98, 1.99) So we can predict a iting function value of 12 9/2/08 Mr. Santowski - Calculus 10 (A) Introduction to Limits We will explore the it in a variety of ways: now using a graph and tracing the function values 9/2/08 Mr. Santowski - Calculus 11 (A) Introduction to Limits Now we can use the TI-89 to actually calculate the it value for us So we have the confirmation of the iting function value of 12 as we had previously with the table and the graph 9/2/08 Mr. Santowski - Calculus 12 4

5 (B) Determining Values of Limits Now, how does all the algebra tie into its? If we try a direct substitution to evaluate the it value, we get 0/0 which is indeterminate g(x) x 2 x 3 8 = x 2 x 2 (2) 3 8 = x 2 (2) 2 = 0 0 9/2/08 Mr. Santowski - Calculus 13 (B) Determining Values of Limits Is there some way that we can use our algebra skills to come to the same answer? Four skills become important initially: (1) factoring & simplifying, (2) rationalizing and (3) common denominators and (4) basic function knowledge 9/2/08 Mr. Santowski - Calculus 14 (B) Determining Values of Limits Consider the expression g(x) = x 3 8 x 2 Now can we factor a difference of cubes? g(x) = x 3 8 x 2 g(x) = (x 2)(x 2 + 2x + 4) (x 2) g(x) = x 2 + 2x + 4, x 2 9/2/08 Mr. Santowski - Calculus 15 5

6 (B) Determining Values of Limits So then, g(x) x 2 x 3 8 = x 2 x 2 = x 2 + 2x + 4 x 2 = (2) 2 + 2(2) + 4 =12 9/2/08 Mr. Santowski - Calculus 16 (B) Determining Values of Limits Determine the following its. Each solution introduces a different algebra trick for simplifying the rational expressions Verify it on GDC x 9 x 9 x 3 x 2 h 0 h 0 1 x 1 2 x 2 (4 + h) 3 64 h 1 (2 + h) h 9/2/08 Mr. Santowski - Calculus 17 x Find x 3 x 3 So we try to use some algebra tricks as before, but x doesnʼt factor. So we use a ToV, and a graph What is the it in this case? 9/2/08 Mr. Santowski - Calculus 18 6

7 In this case, both the graph and the table suggest two things: (1) as x 3 from the left, g(x) becomes more and more negative (2) as x 3 from the right, g(x) becomes more and more positive 9/2/08 Mr. Santowski - Calculus 19 So we write these ideas as: x x 3 x 3 = & x x 3 + x 3 = Since there is no real number that g(x) approaches, we simply say that this it does not exist 9/2/08 Mr. Santowski - Calculus 20 Now here is a graph of a function which is defined as 2 x x < 2 f (x) = (x 3) 2 2 x 2 Find x 2 f(x) 9/2/08 Mr. Santowski - Calculus 21 7

8 Now find the it of this function as x approaches 2 where f(x) is defined as 4 3x x < 2 f (x) = (2x 3) 2 2 x 2 i.e. determine x 2 f(x) 9/2/08 Mr. Santowski - Calculus 22 In considering several of our previous examples, we see the idea of one and two sided its. A one sided it can be a left handed it notated as f ( x) which means we approach x = a from the left x a (or negative) side We also have right handed its which are notated as f ( x ) which means we approach x = a from the right + x a (or positive) side 9/2/08 Mr. Santowski - Calculus 23 We can make use of the left and right handed its and now define conditions under which we say a function does not have a iting y value at a given x value ==> by again considering our various examples above, we can see that some of our functions do not have a iting y value because as we approach the x value from the right and from the left, we do not reach the same iting y value. Therefore, if f ( x) f ( x) then f ( x) does not + exist. x a x a x a 9/2/08 Mr. Santowski - Calculus 24 8

9 (D) Limits at Infinity Consider the following it(s): f (x) if x f (x) = 1 x n Determine the iting value(s) if possible. 9/2/08 Mr. Santowski - Calculus 25 (D) Limits at Infinity In considering its at infinity, we are being asked to make our x values infinitely large and thereby consider the end behaviour of a function 2 x 1 Consider the it x ± x algebraically numerically, graphically and We can generate a table of values and a graph (see next slide) So here the function approaches a iting value, as we make our x values sufficiently large we see that f(x) approaches a iting value of 1 in other words, a horizontal asymptote 9/2/08 Mr. Santowski - Calculus 26 (D) Limits at Infinity Graph & Table 9/2/08 Mr. Santowski - Calculus 27 9

10 (D) Limits at Infinity Algebra x 2 1 x + x 2 +1 x 2 = x 1 2 x 2 x + x 2 x x = x 2 x x 2 = =1 Divide through by the highest power of x Simplify Substitute x = 1/ 0 9/2/08 Mr. Santowski - Calculus 28 (D) Examples of Limits at Infinity Work through the following examples graphically, numerically or algebraically 3x 2 x 2 (i) x 5x 2 + 4x +1 3x 4 x 2 (ii) x 5x 2 + 4x +1 3x 2 x 2 (iii) x 5x 4 + 4x +1 Work through the following examples graphically, numerically or algebraically x tan 1 x ( x x) x ( ( )) 9/2/08 Mr. Santowski - Calculus 29 (E) Limit Laws The it of a constant function is the constant The it of a sum is the sum of the its The it of a difference is the difference of the its The it of a constant times a function is the constant times the it of the function The it of a product is the product of the its The it of a quotient is the quotient of the its (if the it of the denominator is not 0) The it of a power is the power of the it The it of a root is the root of the it 9/2/08 Mr. Santowski - Calculus 30 10

11 (E) Limit Laws Here is a summary of some important its laws: (a) sum/difference rule [f(x) + g(x)] = f(x) + g(x) (b) product rule [f(x) g(x)] = f(x) g(x) (c) quotient rule [f(x) g(x)] = f(x) g(x) (d) constant multiple rule [kf(x)] = k f(x) (e) constant rule (k) = k These its laws are easy to work with, especially when we have rather straight forward polynomial functions 9/2/08 Mr. Santowski - Calculus 31 (E) Limit Laws - Examples Find x 2 (3x 3 4x x 5) using the it laws x 2 (3x 3 4x x 5) = 3 x 2 (x 3 ) 4 x 2 (x 2 ) + 11 x 2 (x) - x 2 (5) = 3(8) 4(4) + 11(2) 5 (using simple substitution or use GDC) = 25 For the rational function f(x), find x 2 (2x 2 x) / (0.5x 3 x 2 + 1) = [2 x 2 (x 2 ) - x 2 (x)] / [0.5 x 2 (x 3 ) - x 2 (x 2 ) + x 2 (1)] = (8 2) / ( ) = 6 9/2/08 Mr. Santowski - Calculus 32 (E) Limit Laws and Graphs 9/2/08 Mr. Santowski - Calculus 33 11

12 (E) Limit Laws and Graphs From the graph on this or the previous page, determine the following its: (1) x -2 [f(x) + g(x)] (2) x -2 [(f(x)) 2 - g(x)] (3) x -2 [f(x) g(x)] (4) x -2 [f(x) g(x)] (5) x 1 [f(x) + 5g(x)] (6) x 1 [ ½f(x) (g(x)) 3 ] (7) x 2 [f(x) g(x)] (8) x 2 [g(x) f(x)] (9) x 3 [f(x) g(x)] 9/2/08 Mr. Santowski - Calculus 34 (F) Applications of Limits The cost of producing a MATH DVD is given by the function C(x) = 15, x where x is the number of DVDʼs produced. The average cost per DVD, denoted by AC(x) is found by dividing C(x) by x. Determine and interpret x AC(x) 9/2/08 Mr. Santowski - Calculus 35 (G) Internet Links Limit Properties - from Paul Dawkins at Lamar University Computing Limits - from Paul Dawkins at Lamar University Limits Theorems from Visual Calculus Exercises in Calculating Limits with solutions from UC Davis 9/2/08 Mr. Santowski - Calculus 36 12

13 (G) Internet Links Limits Involving Infinity from Paul Dawkins at Lamar University Limits Involving Infinity from Visual Calculus Limits at Infinity and Infinite Limits from Pheng Kim Ving Limits and Infinity from SOSMath 9/2/08 Mr. Santowski - Calculus 37 (H) Homework Text, p Q5,7,9,11,13 (graphs) Q15,17 (ToV) Q23,24,25,28,29 (it laws) Q33,37,39,42,45,48,63, (algebra) Q53,58,67 Q74,75,81 (word problems) 9/2/08 Mr. Santowski - Calculus 38 (I) A Level Investigation Research the DELTA-EPSILON definition of a it Tell me what it is and be able to use it MAX 2 page hand written report (plus graphs plus algebra) + 2 Q quiz 9/2/08 Mr. Santowski - Calculus 39 13