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Section 3.5 – Limits at Infinity. Vertical Asymptotes and Limits When we investigated infinite limits and vertical asymptotes, we let x approach a number

Apr 01, 2015

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Section 3.5 Limits at Infinity Slide 2 Vertical Asymptotes and Limits When we investigated infinite limits and vertical asymptotes, we let x approach a number. The result was that the values of y became arbitrarily large (positive or negative). Slide 3 White Board Challenge Analytically find the vertical asymptote(s) of: Slide 4 Horizontal Asymptotes and Limits When we investigate infinite limits and horizontal asymptotes, we will let x become arbitrarily large (positive or negative) and see what happens to y. This will be referred to as the end behavior. Slide 5 End Behavior Let f be a function defined on some interval (a,). Then means that the values of f(x) get closer to L as x increases. Slide 6 End Behavior Let f be a function defined on some interval (-, a). Then means that the values of f(x) get closer to L as x decreases. Slide 7 End Behavior Let f be a function defined on some interval (a,). Then means that the values of f(x) become large (positive or negative) as x increases. Slide 8 End Behavior Let f be a function defined on some interval (-, a). Then means that the values of f(x) become large (positive or negative) as x decreases. Slide 9 White Board Challenge Sketch a graph of a function with the following characteristics: The function is continuous for all reals except 5. Slide 10 Calculating Limits at Infinity Our book focuses on three ways: 1.Numerical Approach Construct a table of values 2.Graphical Approach Draw a graph 3.Analytic Approach Use Algebra or calculus First Second Slide 11 Example 1 Use the graph and complete the table to find the limit (if it exists). x 01510501001000 f(x)f(x) 0 0.9230.980 0.9999 98 0.99980.9992 As x increases, the value of the function approaches 1. Slide 12 Example 2 Use the graph and complete the table to find the limit (if it exists). x -1000-100-50-10-50 f(x)f(x) -3003-303 -153-32.778 -5 UND-17.5 As x decreases, the value of the function decreases. Slide 13 Example 3 Use the graph and complete the table to find the limit (if it exists). x 01510501001000 f(x)f(x) UND5 0.040.005 0.0000 00005 0.0000 05 0.0000 4 As x increases, the value of the function approaches 0. Slide 14 Example 4 Use the graph and complete the table to find the limit (if it exists). x -9999-5000-1000-100-100 f(x)f(x) 0.0010.002 0.010.1 UND 101 As x decreases, the value of the function approaches 0. Slide 15 Special Property of Limits to Infinity If A is any real number and r is a positive rational number then, Furthermore, if r is such that x r is defined for x < 0, then Slide 16 White Board Challenge Use a table or graph to find the limit: Slide 17 Two Procedures for Analytically Determining Infinite Limits If the function is a rational function or a radical/rational function: 1.Divide each term in the numerator and denominator by the highest power of x that occurs in the denominator. 2.Use basic limit laws and the Special Property of Infinite Limits to evaluate the limit. OR Use LHpitals Rule to evaluate the limit (Only if LHpitals Rule applies.) Slide 18 Reminder Slide 19 Example 1 (Procedure 1) Analytically evaluate. In order to use previous results, divide both the numerator and denominator by the highest power of x appearing in the fraction Use Direct Substitution and previous results. Slide 20 ***Aside*** Analytically evaluate. For this example, the limits value does not change if x approaches negative infinity. Slide 21 LHpitals Rule applies since this is an indeterminate form. Example 1 (Procedure 2) In order to use LHpitals Rule direct substitution must result in 0/0 or /. Analytically evaluate Differentiate the numerator and the denominator. Find the limit of the quotient of the derivatives. This is still an indeterminate form, apply LHpitals Rule again to the new limit. Differentiate the new numerator and the denominator. Find the limit of the quotient of the second derivatives. Since the result is finite or infinite, the result is valid. Slide 22 Example 2 (Procedure 1) Analytically evaluate. In order to use previous results, divide both the numerator and denominator by the highest power of x appearing in the fraction Use Direct Substitution and previous results. Slide 23 ***Aside*** Analytically evaluate. For this example, the limits value does not change if x approaches negative infinity. Slide 24 Example 2 (Procedure 2) Analytically evaluate. LHpitals Rule applies since this is an indeterminate form. In order to use LHpitals Rule direct substitution must result in 0/0 or /. Differentiate the numerator and the denominator. Find the limit of the quotient of the derivatives. This is still an indeterminate form, apply LHpitals Rule again to the new limit. Differentiate the new numerator and the denominator. Find the limit of the quotient of the second derivatives. This is still an indeterminate form, apply LHpitals Rule again to the new limit. Differentiate the new numerator and the denominator. Find the limit of the quotient of the third derivatives. Since the result is finite or infinite, the result is valid. Slide 25 Example 3 (Procedure 1) Analytically evaluate. In order to use the previous result, divide both the numerator and denominator by the highest power of x appearing in the fraction Use Direct Substitution and previous results But, in order to simplify the numerator, you must rewrite 1/x Slide 26 ***Aside*** Analytically evaluate. For this example, the limits value does change if x approaches positive infinity. Slide 27 Example 3 (Procedure 2) Analytically evaluate. In order to use LHpitals Rule direct substitution must result in 0/0 or /. Differentiate the numerator and the denominator. LHpitals Rule applies since this is an indeterminate form. Find the limit of the quotient of the derivatives. This is still an indeterminate form, apply LHpitals Rule again to the new limit. Differentiate the new numerator and the denominator. Find the limit of the quotient of the second derivatives. LHpitals Rule has failed to find a limit. This final result is almost identical to the original. The first procedure is more applicable. Slide 28 Example 4 (Procedure 1) Analytically evaluate the following limit: Now evaluate the limit: Since the denominator is not a polynomial, we can not use the first procedure. We need to try something new. Rewrite the expression as a ratio in order to use the first procedure. Strategy: Rewrite one factor so its numerator is 1. Slide 29 Example 4 (Procedure 1) Analytically evaluate the following limit: In order to use LHpitals Rule direct substitution must result in 0/0 or /. Differentiate the numerator and the denominator. Find the limit of the quotient of the derivatives. LHpitals Rule applies since this is an indeterminate form. Since the result is finite or infinite, the result is valid. Rewrite the expression as a ratio in order to use LHpitals Rule. Strategy: Rewrite one factor so its numerator is 1. Slide 30 ***Aside*** Analytically evaluate the following limit: For this example, the limits value does not change if x approaches negative infinity. Slide 31 Day 45: November 10 th Objective: Determine (finite) limits at infinity, horizontal asymptotes of a graph if they exist, and infinite limits at infinity Homework Questions Notes: Section 3.5 Conclusion Homework: Read pgs. 198-204 and complete 3.5 3.5 Slide 32 White Board Challenge Analytically evaluate each limit below: Slide 33 Then y = 1 is a horizontal asymptote. Horizontal Asymptotes and Limits The line y = L is called a horizontal asymptote of the curve y = f(x) if L is finite and either Since: Slide 34 Procedure for Finding Horizontal Asymptotes For a function f : Find the limit of the function as x goes to positive infinity. Find the limit of the function as x goes to negative infinity. If either of the above limits is finite, then they represent a horizontal asymptote(s) (remember to write the result as y = ) Slide 35 Examples Continued For our previous examples: FunctionHorizontal Asymptotes y = 3/5 y = 0 y = 1 NONE Slide 36 Whiteboard Challenge On a calculator, graph What is a characteristic of this graph that we have not discussed? Slide 37 Whiteboard Challenge Slant/Oblique Asymptotes. Slide 38 Oblique/Slant Asymptote For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division. Degree = 2 Degree = 1 Slide 39 Procedure for Finding Oblique/Slant Asymptotes of a Rational Function In a rational function f, if the degree of the numerator is one more than the degree of the denominator: 1.Perform Polynomial division. 2.Ignoring the remainder, the result is the oblique/slant asymptote. (remember to write the result as y = ) Slide 40 Example Analytically find the slant asymptote of x - 3 x x2x2 -3x 2x2x 2 -6 4 Rm Perform Polynomial Division. x 2 x 2 Thus: This means y = x + 2 is a slant asymptote because: Ignore the remainder Slide 41 Asymptotes Summary The following asymptotes exists if Vertical: When there is a non-removable discontinuity (a value for x that makes the denominator 0 and the numerator non-zero) Horizontal: When the limit as x approaches infinity (positive or negative), the value for y approaches a real number. Slant: For a rational function, the degree of the numerator is one more than the degree of the denominator.