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1.5 Notes - Infinite Limits and 3.5 Notes - Limits at Infinity AP Calculus AB Created by Ms. Hernandez Modified by Mr. Harp AP Prep Questions / Warm Up No Calculator! ln x lim x 1 x (a) 1 (b) 0 (c) e (d) –e (e) Nonexistent ( x 2) lim 2 x 2 x 4 (a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE AP Prep Questions / Warm Up No Calculator! ln x ln1 0 lim 0 x 1 x 1 1 (a) 1 (b) 0 (c) e (d) –e (e) Nonexistent ( x 2) ( x 2) 1 1 lim 2 lim lim x 2 x 4 x 2 ( x 2)( x 2) x 2 ( x 2) 4 (a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE Part I: Infinite Limits: 4 3 1 f x x 2 1 -4 -3 -2 -1 0 1 2 3 4 -1 1 lim x 0 x -2 -3 -4 vertical asymptote at x=0. 1 lim x 0 x online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc02_2.ppt IMPORTANT NOTE: f ( x) The equal sign in the statement lim x c does NOT mean the limit exists! “On the contrary, it tells HOW the limit FAILS to exist.” Page 83 Dr. Larson & Dr. Hostetler www.vcsc.k12.in.us/lincoln/math/CaseNotes/.../InfiniteLimits.ppt Definition of a Vertical Asymptote If f(x) approaches infinity or negative infinity as x approaches c from the left or right, then x = c is a vertical asymptote of f. www.vcsc.k12.in.us/lincoln/math/CaseNotes/.../InfiniteLimits.ppt Digging deeper… Infinity is a very special idea. We know we can't reach it, but we can still try to work out the value of functions that have infinity in them. http://www.mathsisfun.com/calculus/limits-infinity.html Question: What is the value of 1/∞ ? Answer: We don't know! Maybe we could say that 1/∞ = 0, ... but if we divide 1 into infinite pieces and they end up 0 each, what happened to the 1? In fact 1/∞ is known to be undefined. http://www.mathsisfun.com/calculus/limits-infinity.html But We Can Approach It! x 1/x 1 1.00000 2 4 10 100 0.50000 0.25000 0.10000 0.01000 1,000 0.00100 10,000 0.00010 http://www.mathsisfun.com/calculus/limits-infinity.html The limit of 1/x as x approaches Infinity is 0 Furthermore: 1 lim n 0, n 0 x x http://www.mathsisfun.com/calculus/limits-infinity.html Limits at Infinity x2 1 lim 2 x x 1 lim x Divide through by the x2 1 2 2 x x x2 1 2 2 x x 1 1 2 x lim x 1 1 2 x highest power of x Simplify 1 0 1 0 1 Substitute 0 for 1/xn www.mrsantowski.com/MCB4U/Notes/PowerPointNotes/BCC016.ppt Example 3x 2 5 x 1 lim x 2 4 x2 Divide by x2 3 5 1 2 x x lim 2 4 x x2 x lim 1 x lim 3 lim 5 x x x lim 4 lim 2 x x 2 2 x 3 0 0 3 04 4 www.rowan.edu/open/depts/math/.../Limits%20and%20Continuity.pp. Determining Infinite Limits from a Graph Example 1 pg 84 Can you get different infinite limits from the left or right of a graph? How do you find the vertical asymptote? Finding Vertical Asymptotes Ex 2 pg 85 Denominator = 0 at x = c AND the numerator is NOT zero Thus, What we have vertical asymptote at x = c happens when both numerator and denominator are BOTH Zero?!?! A Rational Function with Common Factors When both the num. and den. are both zero then we get an indeterminate form and we have to do something else … Ex 3 pg 86 x2 2x 8 lim x 2 x 4 2 Direct sub yields 0/0 or indeterminate form We simplify to find vertical asymptotes but how do we find the limit? When we simplify we still have indeterminate form. x4 lim , x 2 x 2 x 2 A Rational Function with Common Factors Ex 3 pg 86: Direct sub yields 0/0 or indeterminate form. When we simplify we still have indeterminate form and we learn that there is a vertical asymptote at x = -2. Take lim as x-2 from left and right 2 2 x 2x 8 x 2x 8 lim lim 2 x 2 x 2 x 4 x2 4 A Rational Function with Common Factors Ex 3 pg 86: Direct sub yields 0/0 or indeterminate form. When we simplify we still have indeterminate form and we learn that there is a vertical asymptote at x = -2. Take lim as x-2 from left and right x 2x 8 lim 2 x 2 x 4 2 x2 2 x 8 lim 2 x 2 x 4 Take values close to –2 from the right and values close to –2 from the left … Make a table and you will see values approaching positive or negative infinity. Determining Infinite Limits Ex 4 pg 86 Denominator = 0 when x = 1 AND the numerator is NOT zero Thus, we have vertical asymptote at x=1 is the limit +infinity or –infinity? Let x = small values close to c Use your calculator to make sure – but they are not always your best friend! But Properties of Infinite Limits Page 87 lim f ( x) x c lim g ( x ) L xc Sum/difference Product L>0, L<0 Quotient (Finite#/infinity = 0) Same properties for lim f ( x ) x c Ex 5 pg 87 Definition of a Horizontal Asymptote If f(x) approaches a finite limit L as x approaches infinity or negeative infinity, then y = L is a horizontal asymptote of f. www.vcsc.k12.in.us/lincoln/math/CaseNotes/.../InfiniteLimits.ppt Asymptotes & Limits at Infinity For the function (a) lim f ( x ) 2x 1 f ( x) , find x x (b) lim f ( x) x (c) lim f ( x) x 0 lim f ( x ) (d) x 0 (e) All horizontal asymptotes (f) All vertical asymptotes Asymptotes & Limits at Infinity 2x 1 f ( x) x For x>0, |x|=x (or my x-values are positive) 2x 1 2 x 1 1 lim f ( x) lim lim lim 2 2 x x x x x x x 1/big = little and 1/little = big sign of denominator leads answer For x<0 |x|=-x (or my x-values are negative) 2x 1 2x 1 1 lim f ( x) lim lim lim 2 2 x x x x x x x y=2 and y=–2 are HORIZONTAL Asymptotes Asymptotes & Limits at Infinity 2x 1 f ( x) x 2x 1 2x 1 1 lim f ( x) lim lim lim 2 x 0 x 0 x 0 x 0 x x x 1 1 2 2 2 lim DNE x little 2x 1 2x 1 1 lim f ( x) lim lim lim 2 2 x x x x x x x 1 1 2 2 2 lim DNE x little 3.5 Limit at Infinity Horizontal asymptotes! Finite#/infinity = 0 Infinity/infinity Divide the numerator & denominator by a denominator degree of x Some examples Ex 2-3 on pages #199-200 What’s the graph look like on Ex 3.c Called KNOW oblique asymptotes (not in calc AB) Guidelines on page 201 Additional example: http://www.youtube.com/watch?v=J0MoB 8aPvXI 2 horizontal asymptotes Ex 4 pg 202 Is the method for solving lim of f(x) with 2 horizontal asymptotes any different than if the f(x) only had 1 horizontal asymptotes? Limits involving Trig functions Ex 5 pg 203 What is the difference in the behaviors of the two trig functions in this example? Oscillating toward no value vs oscillating toward a value Word Problems !!!!! Taking information from a word problem and apply properties of limits at infinity to solve Ex 6 pg 203 A word on infinite limits at infinity Sometimes, when we determine a limit of a f(x) as x->∞, it comes out ∞ or - ∞ Ex 7 on page 204 Uses Ex property of f(x) 8 on page 204 Uses LONG division of polynomials-Yuck! The shortcut for limits AT infinity Whenever dealing with evaluating limits of the forms () () lim , or lim , →∞ () →−∞ () where P(x) and Q(x) are both polynomials, you can ignore all but the highest degree term in the numerator and the highest degree term in the denominator! This is because as x approaches ∞ or -∞, the highest degree term dominates. The next 5 slides go through this in detail. Limits at Infinity of Polynomial Functions What about limits at infinity for polynomial functions? As x increases without bound in either the positive or the negative direction, the behavior of the polynomial graph will be determined by the behavior of the leading term (the highest degree term). The leading term will either become very large in the positive sense or in the negative sense (assuming that the polynomial has degree at least 1). In the first case the function will approach and in the second case the function will approach -. In mathematical shorthand, we write this as lim f ( x ) x This covers all possibilities. Barnett/Ziegler/Byleen Business Calculus 11e 31 Limits at Infinity and Horizontal Asymptotes A line y = b is a horizontal asymptote for the graph of y = f (x) if f (x) approaches b as either x increases without bound or decreases without bound. Symbolically, y = b is a horizontal asymptote if lim f ( x) b or x lim f ( x) b x In the first case, the graph of f will be close to the horizontal line y = b for large (in absolute value) negative x. In the second case, the graph will be close to the horizontal line y = b for large positive x. Note: It is enough if one of these conditions is satisfied, but frequently they both are. Barnett/Ziegler/Byleen Business Calculus 11e 32 Example This figure shows the graph of a function with two horizontal asymptotes, y = 1 and y = -1. Barnett/Ziegler/Byleen Business Calculus 11e 33 Horizontal Asymptotes SPECIFICALLY for Rational Functions If am x m am1 x m1 a1 x a0 f ( x) , am 0, bn 0 n n 1 bn x bn1 x b1 x b0 am x m lim f ( x) lim then x x b x n n There are three possible cases for these limits. 1. If m < n, then lim f ( x) 0 x The line y = 0 (x axis) is a horizontal asymptote for f (x). am f ( x) 2. If m = n, then xlim bm The line y = am/bn is a horizontal asymptote for f (x) . 3. If m > n, f (x) does not have a horizontal asymptote. Barnett/Ziegler/Byleen Business Calculus 11e 34 Horizontal Asymptotes of Rational Functions (continued) Notice that in cases 1 and 2 on the previous slide that the limit is the same if x approaches or -. Thus a rational function can have at most one horizontal asymptote. (See figure). Notice that the numerator and denominator have the same degree in this example, so the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. y 3x 5 x 9 2x2 7 2 Barnett/Ziegler/Byleen Business Calculus 11e y = 1.5 35