Limits and Asymptotes

(1) Definition (Infinite Limit) Let be a function defined on both sides of except possible at it self. Then



means that the values of can be made arbitrarily large by taking sufficiently close to   ( ).

(2) Definition (Infinite Negative Limit) Let be a function defined on both sides of except possible at itself. Then



means that the values of can be made arbitrarily large negative by taking sufficiently close to ( ).

(3) Proposition (Infinite Limit Theorem) Let be a positive real number.

    (i) If is a positive even integer, then



    (ii) If is a positive odd integer, then

(4) Example (Infinite Limit) Evaluate the limit,  

    Solution. Notice that increases without bound as and therefore,

(5) Definition (Vertical Asymptote) The line is called a vertical asymptote of the curve if at least one of the following statements is true:

(6) Example (Vertical Asymptote) Determine the vertical asymptotes of the functions,

    Solution. Since,



and therefore, the vertical asymptotes are and because,



(7) Example (Vertical Asymptote) Determine the vertical asymptotes of the functions,

    Solution. The function is continuous on its domain which is and therefore, there are no vertical asymptotes for this function.

(8) Example (Vertical Asymptote) Determine the vertical asymptotes of the functions,

    Solution. We can write this function has
    


Therefore, the zeros of the sine and cosine functions yield the vertical asymptotes of for all integers The graph is

(9) 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

  

(10) 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

(11) 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.

(12) 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

(13) 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,
    










(14) Example (Evaluating Limits to Infinity) Evaluate the limit,  

    Solution. We use the conjugate radical as follows,

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

   or   

(16) 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.    

(17) 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




(18) 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 And Asymptotes
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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