Limits to Infinity and Horizontal Asymptotes

    In mathematics, the symbol is not a number, but it is used to describe the process of unrestricted growth or the result of such a growth.  

Definition (Limit to Infinity) Let be a function defined on some interval Then



means that the values of can be made arbitrarily close to by taking sufficiently large; or more precisely, for every there exists an such that

  

Definition (Limit to Negative Infinity) Let be a function defined on some interval Then



means that the values of can be made arbitrarily close to by taking sufficiently large negative; or more precisely, for every there exists an such that

  

    The following limit rules are similar to the limit rules used for when but instead use and they are also valid for when is replaced by

Proposition (Limits to Infinity Rules) If and exist, then

    (i) (Constant) for any constant
        
    (ii) (Multiple)
    
    (iii) (Sum)
    
    (iv) (Difference)
    
    (v) (Product)
    
    (vi) (Quotient)
    
    (vii) (Power)   where is a rational number and whenever the limits exist.
    

Proposition (Limits to Infinity Theorem) Let be a real number.

    (i) If is a rational number, then



    (ii) If is a rational number such that is defined for all then

Example (Evaluating Limits to Infinity) Evaluate the limit,

    Solution. To evaluate the limit we divide both the numerator and the denominator by the highest power of that occurs. So we have,
    










Example (Evaluating Limits to Infinity) Evaluate the limit,  

    Solution. We use the conjugate radical as follows,

    Limits to infinity are very useful for curve sketching; in particular for determine the long term behavior of a function, as in horizontal asymptotes.

Definition (Horizontal Asymptote) The line is called a horizontal asymptote of the curve if either

   or   

Proposition (Horizontal Asymptote Theorem) If



where is the degree of the polynomial in the numerator and is the degree of the polynomial in the denominator, then the horizontal asymptote of the curve determined by the following.

    (i) If then is the horizontal asymptote.
    
    (ii) If then is the horizontal asymptote.
        
    (iii) If then there is no horizontal asymptote, but rather a slant (oblique) asymptote and can be found be using long division.
    

Example (Finding Horizontal Asymptotes) Find the horizontal asymptote of the graph of the function



    Solution. The degree of the numerator is 2 and the degree of the denominator is 2 and therefore we use the leading coefficients to obtain the horizontal asymptote of



Example (Finding Horizontal Asymptotes) Find the horizontal asymptote of the graph of the function



    Solution. Dividing both numerator and denominator by and using the properties of limits, we have
    










Therefore, the line is a horizontal asymptote. It is also important to realize that











Therefore, the line is another horizontal asymptote.

Example (Finding Horizontal Asymptotes) Find the horizontal asymptote of the graph of the function



    Solution. Dividing both numerator and denominator by and using the properties of limits, we have
    














Therefore, the line is a horizontal asymptote. In computing the limit we must remember that for we have so when we divide the numerator by when we have,











Therefore, the  horizontal asymptotes are

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Cite this as:
Limits To Infinity And Horizontal Asymptotes
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/limits-to-infinity-and-horizontal-asymptotes.html