How to Find Horizontal Asymptotes

What is a Horizontal Asymptote

An asymptote is a line or curve that become arbitrarily close to a given curve. In another words it is a line close to a given curve, such that the distance between the curve and the line approaches zero when the curve is reaching higher/ lower values. The region of the curve that has an asymptote is asymptotic. Asymptotes are often found in rotational functions, exponential function and logarithmic functions. Asymptote parallel to the x-axis is known as a horizontal axis. 

How to Find the Horizontal Asymptote

An asymptote exists if the function of a curve is satisfying following condition. If f(x) is the curve, then a horizontal asymptote exist if,

How to find horizontal asymptotes | clickspay.ru

Then horizontal asymptotes exist with equationy=C. If the function approaches finite value (C)at infinity, the function has an asymptote at that valueand the equation of an asymptote is y=C. A curve may intersect this line at several points, but becomes asymptotic as it approaches infinity.

To find the asymptote of a given function, find the limits at infinity.

Finding horizontal asymptotes – Examples

  • Exponential functions of form f(x)=ax and [a>0]

How to find horizontal asymptotes | clickspay.ru

Exponential functions are the simplest examples of horizontal asymptotes.

Taking the limits of the function at positive and negative infinities gives, limx→-∞ ax =+∞ and limx→-∞ ax =0. The right limit is not a finite number and tends to positive infinity, but the left limit approaches the finite values 0.

Therefore, we can say that exponential function f(x)=ax has a horizontal asymptote at 0. The equation of the asymptote line is y=0, which also is the x-axis. Since a is any positive number, we can consider this as a general result.

When a=e= 2.718281828, the function is also known as the exponential function. f(x)=ex has specific characteristics and therefore, important in mathematics.

How to find horizontal asymptotes | clickspay.ru

  • Rational functions

A function of the form f(x)=h(x)/g(x) where h(x),g(x) are polynomials and g(x)≠0, is known as a rational function. Rational function may have both vertical and horizontal asymptotes.

i. Consider the function f(x)=1/x

Function f(x)=1/x has both vertical and horizontal asymptotes.  How to find horizontal asymptotes | clickspay.ru
To find the horizontal asymptote find the limits at infinity.
 limx→=+∞ 1/x=0and limx→=-∞ 1/x =0
When x→+∞, function approaches 0 from the positive side and when x→=-∞ function approaches 0 from the negative direction.
Since function has a finite value 0 when approaching infinities, we can deduce that the asymptote is y=0.

ii. Consider the function f(x)=4x/(x2+1)

How to find horizontal asymptotes | clickspay.ru

 

 

 

 

 

 

 

Again find the limits at infinity to determine the horizontal asymptote.

How to find horizontal asymptotes | clickspay.ru

Again the function has asymptote y=0, also in this case the function intersects the asymptote line at x=0

iii. Consider the function f(x)=(5x2+1)/(x2+1)

How to find horizontal asymptotes | clickspay.ru

Taking the limits at infinity gives,

How to find horizontal asymptotes | clickspay.ru

Therefore, the function has finite limits at 5. So, the asymptote is y=5

About the Author: admin

Leave a Reply


Related pages


graveyard vs cemeterymolasses versus blackstrap molasseschloroplast membrane functiondifference between mercy and gracetradeoff and opportunity costcauses leukopeniadefinition of bewilderedcentripetal force equationpure compound definition chemistryexamples of polysemy in englishverbal irony definition in literaturewhat is the difference between sparkling wine and champagnefolktale vs fairy talewhat are the difference between formal and informal letterthermosetting plastics listconductor and insulator differencedefinition of atomic orbitaldifference between crepe and pancakedifference between emotional and sentimentaloverhead cost per unit activity based costingrough endoplasmic reticulum functiondifference between salmonella typhi and salmonella typhimuriumdifference between syllabus and course outlinedefinition consistantsuffix and affixwhat is the difference between de jure and de factofrance popular foodjuxtaposition poetry definitioncomparison between photosynthesis and cellular respirationwhat is an example of acculturationmolarity vs densityhow to analyze a poemdiff between alligator and crocodilesugarandhoneydog breed doxinadjective of quality definitiondiurnal nocturnalbicameral legislaturewhat is the difference between relief and elevationwhat is the difference between cellulose and starchwhat is the difference between a comedy and a tragedyis patience a noundidactic in a sentenceromanticism and neoclassicismdifference between pit bullswhat is a infinitive phrasedefine aneuploidylipids physical propertiessexual and asexual reproduction definitiongold foil experiment explainedpast tense for buildwhat is norepinephrine functiondefinition of sporophytedifference between tutor and teacherwhat is the difference between a protagonist and antagonistdifference mitosis meiosiscave of ajantacauses of diminishing returnscharacteristics of unicellular and multicellular organismsdifference between diabetes and diabetes mellitustransnational firmsproperties of paramagnetic materialswhat influenced art nouveaudifference between neoplasm and tumordifference between associate and bachelor degreeamerican vs german rottweilerseverybody vs everyone grammarfull size dachshundbravery vs couragewhat is so42immoral vs amoraldifference between analog multimeter and digital multimetersour and bitter foodsbasophils eosinophilsenunciate meansthe good samaritan moral