How do you Find the Horizontal Asymptotes of a Function

Problems concerning horizontal asymptotes appear on both the AP Calculus AB and BC exam, and its important to know how to find horizontal asymptotes both graphically (from the graph itself) and analytically (from the equation for a function). Before we delve into finding the asymptotes though we better see what exactly an asymptote is. Definition of Horizontal Asymptote A horizontal asymptote for a function is a horizontal line that the graph of the function approaches as x approaches ∞ (infinity) or -∞ (minus infinity). In other words, if y = k is a horizontal asymptote for the function y = f(x), then the values (y-coordinates) of f(x) get closer and closer to k as you trace the curve to the right (x→ ∞) or to the left (x → -∞). The Limit Definition for Horizontal Asymptotes Because asymptotes are defined in this way, it should come as no surprise that limits make an appearance.   The precise definition of a horizontal asymptote goes as follows:   We say that  y =  k is a horizontal asymptote  for the function y = f(x) if either of the two limit statements are true:  . Finding Horizontal Asymptotes Graphically A function can have two, one, or no asymptotes. For example, the graph shown below has two horizontal asymptotes, y = 2 (as x→ -∞), and y = -3 (as x→ ∞). If a graph is given, then simply look at the left side and the right side. If it appears that the curve levels off, then just locate the y-coordinate to which the curve seems to be approaching. It helps to sketch a horizontal line at the height where you think the asymptote should be. Lets see how this works in the next example. Keep in mind, you will typically not be shown the dashed line — that would make the problem way too easy! The graph on the left shows a typical function. If you follow the left part of the curve as far to the left as you can, where do you end up? In other words, what is the y-coordinate of the leftmost point shown in the graph? A good estimate might be somewhere between 1 and 2, perhaps a little closer to 1. Well imagine what would happen if you continued drawing the graph to the left of what is shown. It seems reasonable that the curve levels off and approaches a value of 1, gently touching down on the horizontal line y = 1 just like an airplane landing. Similarly, follow the right part of the curve as far to the right as you can, and imagine what would happen if you kept going. Again, the curve seems to level off and approach y = 1, this time coming up from below the line. This function has a single horizontal asymptote, y = 1. Once you sketch the line (dashed in the righthand figure), it becomes clear that we have found the correct horizontal asymptote. Finding Horizontal Asymptotes Analytically What if you are not given a graph? Well in many cases its actually quite easy to determine the horizontal asymptote(s), if any exist. There are just a few rules to follow. Rational Functions If your function is rational, that is, if f(x) has the form of a fraction, f(x) = p(x) / q(x), in which both p(x) and q(x) are polynomials, then you can use highest order term analysis. The highest order term of a polynomial p(x) is the single term having the greatest degree (exponent on x). For example, the highest order term of 6x 3x5 + 5x3 + 42 is: –3x5. Highest Order Term Analysis To do highest order term analysis on a rational function, make sure the top and bottom polynomials are fully expanded and then write a new function having only the highest order term from the top and from the bottom. All other terms (lower order terms) can safely be ignored. Cancel any common factors and variables and: If the result is a constant k, then y = k is the single horizontal asymptote. This happens when the degree of the top matches the degree of the bottom. If the result has any powers of x left over on top, then there is no horizontal asymptote. If the result has any powers of x left over on bottom, then y = 0 is the single horizontal asymptote. Examples for Highest Order Term Analysis Lets use highest order term analysis to find the horizontal asymptotes of the following functions. (a) The highest order term on the top is 6x2, and on the bottom, 3x2. Dividing and cancelling, we get (6x2)/(3x2) = 2, a constant. Therefore the horizontal asymptote is y = 2. (b) Highest order term analysis leads to (3x3)/(x5) = 3/x2, and since there are powers of x left over on the bottom, the horizontal asymptote is automatically y = 0. (c) This time, there are no horizontal asymptotes because (x4)/(x3) = x/1, leaving an x on the top of the fraction. Exponential Functions The method of highest order term analysis is quick and easy but only applies to rational functions. What if you are given a different kind of function? Certain functions, such as exponential functions, always have a horizontal asymptote. A function of the form f(x) = a (bx) + c always has a horizontal asymptote at y = c. For example, the horizontal asymptote of y = 30e6x – 4 is: y = -4, and the horizontal asymptote of y = 5 (2x) is y = 0. Horizontal Asymptotes in General? More general functions may be harder to crack. However, just remember that a horizontal asymptote are technically limits (as x→ ∞ or x→ -∞). Therefore, they measure the end behavior of the function. If you are working on a section of the exam that allows a graphing calculator, then you may simply graph the function and trace it to the right and left until you can determine whether the values level off in either direction. Conclusion Problems about horizontal asymptotes are usually not too difficult. Know how to look at the graph, or if a graph is not given, then know how to analyze the function (highest order term analysis for rational functions, the special rule for exponential functions, or when all else fails, try graphing).

Victim or Traitor The Judas Theme in Arnows The Dollmaker - Literature Essay Samples

The legend of Judas is a constant background murmur in Harriette Arnows The Dollmaker. It begs us to wonder: is Gertie Nevels a victim or a betrayer? Many believe that Christs betrayal was preordained and that Judas, with his kiss, was obeying Gods command, suggesting his innocence of wrongdoing. Others believe that his act was a willful sin. To a lesser degree, Gertie can also be seen as a traitor, betraying herself and her children as she tries to obey her mother and the Bibles command, Wives, be in subjection unto your husbands (154). Throughout the story, she both identifies with and pities Judas. Gerties preoccupation with Judas is revealed early on when she remarks that she would like to carve the traitors likeness. Soon after, were introduced to Gerties block of wild cherry wood, with its man hidden deep inside and only the top of his head showing. Gertie believes that one day shell bring him out a that block (48), hoping that the man will be Christ, but secretly believing hel l be Judas the Judas she had pitied (139). When she finds she can buy the Tipton place and fulfill her dreams, her joy is expressed in her sudden belief that it was Christ in the block of wood after all (138). However, her first act of betrayal occurs shortly thereafter, when she lets her mother shame her into following Clovis to Detroit. Despite the strength of character she displays on other occasions, she crumbles in the face of her mothers vitriol, losing her oldest sons respect and effectively condemning herself and the children to a type of hell. As her dream of owning land crumbles away, she thinks, she had always known that Christ would never come out of the cherry wood (158). Later, on the train to Detroit, she struggles with guilt over the expense of bringing the block, thinking, Judas wood it seemed now (164). Her feelings toward the wood reflect her own unarticulated feelings about herself. She knows she has betrayed herself by giving up her dreams.In Detroit, Gertie co ntinues to identify with Judas. After saving the gospel woman from Mrs. Dalys broom, Gertie is troubled by the rage in Mrs. Dalys eyes. She blames herself for it: A sin it was to make another sin with such hatred and such talk, but Judas had to sin (226). Soon after, she betrays her oldest son by humiliating him in front of Mr. Daly. When Clovis insists, its you thats as much wrong with Reuben as anything (380), she betrays Reuben again by counseling him to try harder to be like the rest (382), telling him (from his perspective, at least) that she doesnt accept who he is. After he runs away, she dwells on Matthew 27:4: I have betrayed innocent blood (382). Obsessively reading the remorseful words of Judas is the closest Gertie can come to articulating her regret. As much as she suffers from the loss of her son, Gertie goes on to betray Cassie in a very similar way. Once again, she ignores her own better judgment and follows Cloviss injunction to make her quit them foolish runnen an talken-to-herself fits (412). She breaks Cassies heart by telling her, harshly, that her imaginary friend doesnt exist there aint no Callie Lou (426) even as she fights the urge to seize and hug and kiss the child, and cry, Keep her, Cassie. Keep Callie Lou (426). Like Reuben, Cassie must deny who she is to be accepted in Detroit, but unlike Reuben, she cant escape this fate by running away. Yet Gertie loses her too when shes killed by a train. Later, Gertie bleakly tells Mrs. Anderson, we all sell our own (496).In some ways, Gertie is a victim, manipulated by bullying and bad advice. Yet she cant be considered innocent. Her actions in Kentucky show that she is a strong and competent woman, capable of making good decisions. By later choosing not to assert herself, she causes harm and is responsible for the consequences. As she pours herself into the cherry block and must finally acknowledge that Christ is not emerging, she is confronted with the truth that she is as much to blame for the familys suffering as Clovis, as her mother, as the war. She faces her conscience as she admits to herself that the wood was Judas after all (585). We remember Cassies voice back in Kentucky, laughingly insisting that the man in the wood is a her (47). The traitors face, never revealed, is perhaps Gerties own.