UNIT 3 Recall From Unit Rational Functions f() is a rational function if where p() and q() are and. Rational functions often approach for values of. Rational Functions are not graphs There various types of discontinuities. There are which occur when only the denominator (bottom) is zero. There are in the graph when there is zero/zero P a g e
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) P a g e
Horizontal Asymptotes The line is a orizontal symptote ( ) of the curve y = f() if either: 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 Graph y sin y What is the HA? NOTE: and This means that if we do direct substitution in a limit we can now determine the following: Note: are all considered to be 3 P a g e
In order to find limits at infinity (HA) we use the previous information: Eamples:. lim 3 The long painful, algebraic, way to solve these limits is to divide each term by the. lim 4 P a g e
3. lim Really, all we need to do is to look at the of the degree terms ( coefficients) The degree term(s) the epression when the value of gets really Solve the following: 4. lim 3 3 5. lim 5 3 4 6. lim 7. lim 6 8. lim 5 P a g e
9. lim 4 0. lim. lim 3. lim 3 3. lim 3 4 4 3 6 P a g e
Infinite Limits At Infinity The notation is used to indicate that the values of f() become as becomes. Similar meanings are attached to the following symbols: Eample Find Solution: When becomes large, 3. For instance, In fact, we can make 3 as as we like by taking l. Therefore we can write Similarly, when is large negative, so is 3. Thus These limit statements can also be seen from the graph of y = 3. 7 P a g e
Find the limit 3. lim 5. lim Tet Page 68 ODD numbers #9 7, 30-33 8 P a g e
Infinite Limits OR Vertical Asymptotes For all values, b, that give a in the and a in the the function will have a at Consider the following functions: What are the vertical asymptotes? A) y B) y C) y 3 D) y E) y When does the graph go in the same direction at a vertical asymptote? A) y Sketch these graphs on the calculator B) y C) y D) y 3 4 E) y 3 F) y ( ) 6 G) y ( ) 3 9 P a g e
Theorem A) If n is a positive even integer then lim 0 n lim a ( a ) n lim n 0 lim a ( a ) n B) If n is an odd integer then lim n 0 lim a ( a ) n lim n 0 lim a ( a ) n Evaluate ( i ) lim f ( ) ( ii ) lim f ( ) A) f( ) B) f( ) ( ) 0 P a g e
C) f( ) D) f( ) ( ) 4 E) f( ) 3 Describe the behaviour of f() at its vertical asymptotes and sketch the graph near the asymptotes. A) f ( ) P a g e
B) f ( ) 4 ( )( ) C ) f ( ) 6 P a g e
3 P a g e
Tet Page 67-8 #,, 3-5 4 P a g e
+ lim n 3 Algebraically find the following limits lim n + 3 In mathematics 00, and previous in this course you were eposed to where and where 5 P a g e
Sketching the Graphs of Rational Functions Intercepts How do you find y-intercepts? Find the y-intercept of 4 f( ) 3 4 How do you find -intercepts? Why is only the considered when determining the -intercepts of a rational function? o o 6 P a g e
Ensure you verify that this value for the -intercept does not result in a denominator of zero. WHY? o which means you must do a to determine whether the function has at that value. Find the -intercept of 4 f( ) 3 4 Hint: Looking at the we have possible -intercepts However would make the denominator equal zero. So the only intercept is: 7 P a g e
So, what is happening when = 4? To determine this we need to find the. What are the asymptotes for this graph? Vertical Asymptotes ( 4) f( ) ( )( 4) There are that give but we just discovered that is a. This leaves We need to check 8 P a g e
Horizontal Asymptotes We need to find. Graph 4 f( ) 3 4 y 9 P a g e
Sketch the following by finding all intercepts and asymptotes Remember to first simplify the rational epression by the numerator and denominator. A factor of corresponds to a. A factor of corresponds to a. When the numerator and denominator of the original function contain a of the 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 if the power is in the Or a if the power is in the. 0 P a g e
4 A) f ( ) 4 P a g e
3 B) f ( ) 5 4 y P a g e
8 C ) f ( ) ( 3) y 3 P a g e
Slant ( or Oblique) Asymptotes These occur when the degree of the numerator is one 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 the numerator by the denominator. Numerical Eample: Find the slant asymptote for the previous problem 8 C ) f ( ) ( 3) 4 P a g e
Find the slant asymptote for 3 4 6 f( ) 4 5 P a g e
Sketch the following: What is the horizontal asymptote? What is the domain? Is the function continuous? Construct a table of values using the following -values and then sketch its graph. y 6 P a g e
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 7 P a g e
B) y 8 P a g e
Sketch of graph of y = f() given: f( ) f(4) 0 lim f ( ) lim f ( ) 0 0 lim f ( ) 0 lim f ( ) 0 y B) Sketch of graph of y = f() given: f( 6) 0 f(0) lim f( ) 3 lim f ( ) lim f ( ) lim f ( ) lim f ( ) y 9 P a g e