Limits in Calculus

(1) Example (Limit Does Not Exist Left and Right Behavior) Find for

.

    Solution. The limit does not exist because



which can be seen from the graph:

(2) Example (Limit Does Not Exist Oscillating Behavior) Find

    Solution. The limit does not exist because   and   for

  and   

respectively. The graph of   is oscillating around so we infer that the limit does not exist because does not approach a number, but rather oscillates, as approaches 0. This behavior can be seen in a good sketch of the graph:

(3) Example (Infinite Limits) Determine

    Solution. Since decreases without bound as and   increases without bound as we say that does not exist.

(4) Proposition (Special Trigonometric Limits) The following trigonometric limits hold:

(5) Example (Special Trigonometric Limits) Find the limit:

    Solution. We have
    


(6) Example (Special Trigonometric Limits) Find the limit:

    Solution. We have

(7) Example (Special Trigonometric Limits) Find the limit:

    Solution. We have







(using and since as we infer that so we continue)

(8) 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   ( ).

(9) 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 ( ).

(10) 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

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

    Solution. Notice that increases without bound as and therefore,

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

    Solution. Note that and decreases without bound as and therefore, This is true because when is close to we know that is negative. Similarily,   because when and close to we know that is negative.

(13) 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

  

(14) 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

(15) 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.
    

(16) 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

(17) 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,
    










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

    Solution. We use the conjugate radical as follows,

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Limits In Calculus
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/limits-in-calculus.html