UNIT 3 Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions
Recall From Unit Rational Functions f() is a rational function if f( ) p ( ) q ( ) where p() and q() are polynomials and q ( ) 0 Rational functions often approach either slant or horizontal asymptotes for large (or small) values of
Rational Functions are not continuous graphs. There various types of discontinuities. There are vertical asymptotes which occur when only the denominator (bottom) is zero. There are holes in the graph when there is zero/zero 0 0
Limits at Infinity and Infinite Limits Horizontal and Vertical Limits
Use graphing technology to graph the following: For large values of, what is the function approaching? A) y 3 B) y C) y
D) E) F) y y y 3 G) H) y y 4 n
Horizontal Asymptotes The line y = L is a Horizontal Asymptote (HA) of the curve y = f() if either: lim f ( ) L OR lim f ( ) Note: A function can cross the HA an infinite number of times, but as gets etremely large ( ) or small ( ) the function approaches the Horizontal Asymptote L
Graph y sin What is the HA? y
NOTE: c c lim 0 and lim 0 r This means that if we do direct substitution in a limit we can now determine the following: c 0 Note: 0 c,, c 0 0 0 r 0 c 0, c 0 are all considered to be
In order to find limits at infinity (HA) we use the previous information: Eamples:. lim 3 Direct Substitution This is an indeterminate form Aka More work The long painful, algebraic, way to solve these limits is to divide each term by the highest power of
. lim lim lim 3 3 3 Thus the horizontal asymptote for is y = /3 3 3 3. lim lim lim Thus the horizontal asymptote is Y = /
3. lim lim 0 0 0 0 lim Thus the horizontal asymptote is y = 0
Really, all we need to do is to look at the coefficients of the highest degree terms (leading coefficients) The highest degree term(s) take over the epression when the value of gets really LARGE
4. lim Solve the following: 3 3 5. lim 5 3 4
6. lim 7. lim 6 8. lim
9. lim 4 0. lim. lim 3
. lim 3 3. lim 3 4 4 3
8
Infinite Limits At Infinity The notation lim f( ) is used to indicate that the values of f() become large as becomes large. Similar meanings are attached to the following symbols: lim f( ) lim f( ) lim f( )
Find Solution: Eample When becomes large, 3 also becomes large. For instance, 3 00,000,000 3 0 000 3 000, 000, 000, 000 In fact, we can make 3 as big as we like by taking large enough.
Solution cont d Therefore we can write lim 3 Similarly, when is large negative, so is 3. Thus lim 3 These limit statements can also be seen from the graph of y = 3.
Find the limit. lim 3. lim 5
Tet Page 68 ODD numbers #9 7, 30-33 3
4
Infinite Limits 5
Infinite Limits OR Vertical Asymptotes For all values, b, that give a non-zero constant in the numerator and a zero in the denominator the function will have a vertical asymptote at = b.
Consider the following functions: What are the vertical asymptotes? ) A y ) B y ) 3 C y ) D y ) E y
When does the graph go in the same direction at a vertical asymptote? Sketch these graphs on the calculator A) y B) y C) y D) y 3 4 E) y F) y 3 ( ) G) y 6 ( ) 3
The graph goes in the same direction at a VA when the eponent is even To find out what the function goes to we check the limit
Theorem A) If n is a positive even integer then lim 0 n lim a ( a ) n lim n lim a ( a 0 ) n
Theorem B) If n is an odd integer then lim 0 n lim a ( a ) n lim 0 n lim a ( a ) n
Evaluate ( i ) lim f ( ) A) B) f( ) ( ii ) lim f ( ) f( ) ( )
C) D) f( ) f( ) ( ) 4 E) f( ) 3
Describe the behaviour of f() at its vertical asymptotes and sketch the graph near the vertical asymptotes A) f ( )
Describe the behaviour of f() at its vertical asymptotes and sketch the graph near the vertical asymptotes B) f ( ) 4 ( )( )
C ) f ( ) 6
#,, 3-5 Tet Page 67-8 38
Algebraically find the following limits lim n + 3 lim n + 3 In mathematics 00, and previous in this course you were eposed to 0 where and = < 0 where = 39
40 lim 3 lim 3 lim 3 lim 3 lim 3 0 0
4 lim 3 lim 3 lim since 0, 3 lim 3 lim 3 0 0
Sketching the Graphs of Rational Functions Intercepts Asymptotes Holes 4
Intercepts How do you find y-intercepts? Let = 0 and solve for y (or f()) Find the y-intercept of f( ) 4 3 4 43
Intercepts How do you find -intercepts? Let y = 0 and solve for Why is only the numerator considered when determining the -intercepts of a rational function.? A zero in the denominator is a discontinuity. The function is undefined when there is division by zero. 44
Ensure you verify that this value for the -intercept does not result in a denominator of zero. WHY? Because then we have 0 0 which means you must do a limit to determine whether the function has a hole or a vertical asymptote at that value. 45
Find the -intercept of f( ) Hint: Factor completely first 4 3 4 f( ) 4 3 4 ( 4) ( )( 4) Looking at the numerator we have two possible -intercepts 0 and 4 However would 4 make the denominator equal zero. So the only intercept is: 0 46
So, what is happening when = 4? To determine this we need to find the limit as approaches 4 ( 4) lim 4 ( ) lim 4 ( )( 4) 4 5 Thus there is a hole (Point of Discontinuity) in the graph of y = f() at the point (4,0.8) 47
What are the asymptotes for this graph? Vertical Asymptotes f( ) ( 4) ( )( 4) There are two -values that give division by zero, but we just discovered that = 4 is a hole. This leaves = - as a possible VA 48
We need to check both the left and right side limits for = -. ( 4) ( 4) lim ( )( 4) lim ( )( 4) 49
Horizontal Asymptotes We need to find the limits as approaches lim 4 3 4 lim 4 3 4 50
Graph f( ) 4 3 4 y 5
Sketch the following by finding all intercepts and asymptotes Remember to first simplify the rational epression by factoring the numerator and denominator. A factor of only the numerator corresponds to a -intercept. A factor of only the denominator corresponds to a vertical asymptote. 5
When the numerator and denominator of the original function contain a common factor of the same eponent the graph has a hole in it. (i.e., point of discontinuity) If the common factors do not have the same eponent the discontinuity may be: A hole if the highest power is in the numerator Or a vertical asymptote if the highest power is in the denominator. 53
Sketch the following by finding all intercepts and asymptotes A) f ( ) 4 4
55
Sketch the following by finding all intercepts and asymptotes 3 B) f ( ) 5 4
y
Sketch the following by finding all intercepts and asymptotes C ) f ( ) 8 ( 3)
y
Slant ( or Oblique) Asymptotes These occur when the degree of the numerator is one larger than the degree of the denominator. In this case the function will approach an oblique line as To find the equation of the oblique asymptote, you must divide the numerator by the denominator.
Numerical Eample: 5 73 73 5 3 54 5
Find the slant asymptote for the previous problem C ) f ( ) 8 ( 3) 3 8 8 ( 3) Now take the limit as approaches infinity 8 lim ( 3)
Find the slant asymptote for f( ) 4 6 4 3
Sketch the following What is the horizontal asymptote? What is the domain? Is the function continuous? 65
Construct a table of values using the following -values and then sketch its graph. -0 - -0.5 0 0.5 0 y y 66
Describe the intercepts and the behaviour of the function y = f() around the asymptotes so that a fellow classmate could draw the graph below. A) y 67
y B) 68
Sketch of graph of y = f() given: f( ) f(4) 0 lim f ( ) lim f ( ) 0 0 lim f ( ) 0 lim f ( ) 0 y 69
Sketch of graph of y = f() given: f( 6) 0 f(0) lim f( ) 3 lim f ( ) lim f ( ) lim f ( ) lim f ( ) y 70