How To Find Asymptote Of Exponential Function

How do y'all Find the Horizontal Asymptotes of a Function?

By on January iv, 2017 , UPDATED ON June 11, 2018, in AP

Bug concerning horizontal asymptotes appear on both the AP Calculus AB and BC exam, and it's of import to know how to find horizontal asymptotes both graphically (from the graph itself) and analytically (from the equation for a function).

Earlier nosotros delve into finding the asymptotes though we better run into what exactly an asymptote is.

Definition of Horizontal Asymptote

A horizontal asymptote for a part is a horizontal line that the graph of the role approaches as x approaches ∞ (infinity) or -∞ (minus infinity). In other words, if y = thousand is a horizontal asymptote for the role y = f(10), then the values (y-coordinates) of f(x) get closer and closer to g as you trace the curve to the correct (x→ ∞) or to the left (x → -∞).

The Limit Definition for Horizontal Asymptotes

Because asymptotes are defined in this way, it should come equally no surprise that limits make an advent.  The precise definition of a horizontal asymptote goes as follows:  We say thaty =k is a horizontal asymptote for the part y = f(x) if either of the two limit statements are true:.

Finding Horizontal Asymptotes Graphically

A function tin can take two, one, or no asymptotes. For example, the graph shown below has two horizontal asymptotes, y = 2 (every bit ten→ -∞), and y = -3 (as x→ ∞).

If a graph is given, and so simply look at the left side and the right side. If it appears that the curve levels off, then only locate the y-coordinate to which the curve seems to be approaching. It helps to sketch a horizontal line at the height where you recall the asymptote should be. Let's run into how this works in the side by side case. Proceed in heed, you will typically not exist shown the dashed line — that would make the trouble way besides piece of cake!

The graph on the left shows a typical office. If you lot follow the left part of the curve equally far to the left as y'all tin, where do you end upwardly? In other words, what is the y-coordinate of the leftmost point shown in the graph? A good guess might be somewhere betwixt 1 and ii, perhaps a little closer to ane.

Well imagine what would happen if you continued drawing the graph to the left of what is shown. Information technology seems reasonable that the curve levels off and approaches a value of one, gently touching down on the horizontal line y = i but like an airplane landing.

Similarly, follow the right role of the curve as far to the right as you lot can, and imagine what would happen if you kept going. Over again, the curve seems to level off and approach y = 1, this fourth dimension coming up from below the line. This office has a unmarried horizontal asymptote, y = 1. Once you lot sketch the line (dashed in the righthand figure), it becomes clear that we accept found the correct horizontal asymptote.

Finding Horizontal Asymptotes Analytically

What if you are not given a graph? Well in many cases it's actually quite easy to determine the horizontal asymptote(southward), if any exist. There are but a few rules to follow.

Rational Functions

If your function is rational, that is, if f(10) has the form of a fraction, f(ten) = p(x) / q(ten), in which both p(ten) and q(x) are polynomials, then you lot can use highest lodge term assay. The highest order term of a polynomial p(x) is the single term having the greatest degree (exponent on ten). For instance, the highest club term of 610 – 3x ^v + vten ³ + 42 is: –iiix ⁵.

Highest Lodge Term Analysis

To do highest order term analysis on a rational function, make sure the meridian and bottom polynomials are fully expanded and then write a new function having merely the highest order term from the summit and from the lesser. All other terms (lower gild terms) tin safely be ignored. Cancel whatsoever common factors and variables and:

• If the event is a constant k, and then y = g is the single horizontal asymptote. This happens when the degree of the superlative matches the degree of the lesser.

• If the result has any powers of x left over on top, then there is no horizontal asymptote.

• If the issue has any powers of ten left over on bottom, then y = 0 is the unmarried horizontal asymptote.

Examples for Highest Order Term Analysis

Let's apply highest order term assay to observe the horizontal asymptotes of the following functions.

(a) The highest guild term on the summit is half-dozen10 ², and on the bottom, 310 ². Dividing and cancelling, we get (half dozenx ^two)/(3x ^two) = 2, a constant. Therefore the horizontal asymptote is y = 2.

(b) Highest order term assay leads to (threex ³)/(10 ⁵) = iii/x ^two, and since in that location are powers of x left over on the bottom, the horizontal asymptote is automatically y = 0.

(c) This time, in that location are no horizontal asymptotes because (10 ^four)/(x ³) = x/i, leaving an ten on the top of the fraction.

Exponential Functions

The method of highest guild term analysis is quick and easy but only applies to rational functions. What if y'all are given a dissimilar kind of function? Certain functions, such as exponential functions, e'er have a horizontal asymptote. A function of the form f(x) = a (b^x) + c always has a horizontal asymptote at y = c. For example, the horizontal asymptote of y = 30e ^{– 6x} – 4 is: y = -4, and the horizontal asymptote of y = 5 (ii¹⁰) is y = 0.

Horizontal Asymptotes in General?

More full general functions may exist harder to crack. However, merely call back that a horizontal asymptote are technically limits (as x→ ∞ or x→ -∞). Therefore, they measure the end beliefs of the function. If you lot are working on a department of the exam that allows a graphing calculator, then you lot may simply graph the function and trace information technology to the right and left until you tin determine whether the values level off in either management.

Determination

Problems about horizontal asymptotes are usually non too difficult. Know how to look at the graph, or if a graph is not given, then know how to analyze the function (highest guild term analysis for rational functions, the special dominion for exponential functions, or when all else fails, try graphing).

• 

  Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Get Bucks!!). He received his BA in Mathematics with a small in information science from Oberlin Higher in 2002. In improver, Shaun earned a B. Mus. from the Oberlin Solarium in the same year, with a major in music limerick. Shaun still loves music -- near equally much every bit math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for virtually a decade, and hopes his experience tin can aid you to succeed!

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