Mth141s42 - Section 4.2 - More on Graphs of Rational...

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Section 4.2 - More on Graphs of Rational Functions HORIZONTAL ASYMPTOTES: Note the graph of 1 ( ) f x x = As x gets very large in value, x +∞ (approaches positive infinity), the graph gets closer to (approaches) the line y = 0 (the x-axis). It approaches the line y = 0 asymptotically. (An asymptote is a line a curve approaches infinitely along its length.). The graph also approaches the line y = 0 as x -∞ . The line y=0 is called the horizontal asymptote . And since the graph also vertically approaches the line x=0, it is called the vertical asymptote . We will be graphing rational functions by noting their horizontal and vertical asymptotes. DEFINITION: A rational function ( ) ( ) ( ) g x f x q x = has a horizontal asymptote at y = b if the values of f(x) approach the value b as x becomes increasingly large. Ex. 6 ( ) 2 x f x x = - Note: x can not = 2! Try some values > 2. x 3 4 5 6 8 10 100 1000 f(x) 18 12 10 9 8 7.5 6.1+ 6.012+ So as x gets very large, the function approaches 6 (from slightly above 6). Try some values < 2. x 1 0 -4 -8 -10 -100 -1000 f(x) -6 0 4 4.8 5 5.88+ 5.988+ So as x gets very large negatively, the function approaches 6 (from slightly below 6).
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So from the above definition, the line y = 6 is a horizontal asymptote for this function. The function will approach the line y = 6 on the “ extreme ends ” of the graph. See the graph below. Graph by itself. Graph with H.A. : y = 6 We have something called a “limit” statement to indicate this result. The graph approaches the value 6 as a limit on each end of the graph. The statements look like : lim ( ) 6 x f x →+∞ = and lim ( ) 6 x f x →-∞ = . This says “The limit of the function as x approaches positive infinity is 6.” Also, “The limit of the function as x approaches negative infinity is 6.” Again, this means the graph will approach the horizontal value y=6 on each end of the graph. VERTICAL ASYMPTOTES: A vertical asymptote is a vertical line ( x = ?) that the graph will approach (but not touch!). A vertical asymptote will occur at a value of x that makes ONLY the denominator of the function = 0. Ex.
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Mth141s42 - Section 4.2 - More on Graphs of Rational...

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