Calculus Limits

    In this topic we concentrate not on the formal definition of a limit of a function of one variable but rather give several examples which emphasis algebra and trigonometry techniques to evaluate limits of functions using basic limit laws.

(1) Proposition (Limit) For any real number suppose the functions and both have finite limits at Then

    (i) (Constant) for any constant
    
    (ii) (Limit of )
    
    (iii) (Multiple)
    
    (iv) (Sum)
    
    (v) (Difference)
    
    (vi) (Product)
    
    (vii) (Quotient)

The above limit rules also hold when is replaced by or

    It is important to realize the assumption in the last stated proposition is required. For example, if exists and does not exist we can not say



does not exist because does not exist. Here is a concrete example of this.

(2) Example (Limit) Suppose and find and   

    Solution. Since we know does not exist, but this does not imply anything about nor To find these limits we first find the two one-sided limits,
    
      



and since we can now say the wo-sided limit does not exist. Similarly,
    
      



and therefore,

(3) Example (Limit) Find the limit of at

    Solution. By using several limit rules, we have



(quotient rule)

(sum and difference rules)

(multiple rule)

(product rule)

(limit of rule)



Notice this is the same as evaluating the rational function at   

(4) Proposition (Trigonometric Functions) If is any real number in the domain of the given function, then

(5) Example (Trigonometric Functions)  Compute the limit of at

    Solution.  By using several limit rules, we have

(6) Example (Finding Limits Using Algebra) Compute the limit of at

    Solution.  By using several limit rules, we have

In the previous example notice that we used



Now it is not true that the functions and are the same function because they have different domains. But the above equality is true because is approaching 2, and not equal to 2. So the point is, because when we can indeed say



This is an important part of understanding calculus limits.

(7) Example (Finding Limits Using Algebra) Given and compute the limit at

    Solution. The graph of is



which inspires to try a factor of from obtaining and a graph of is


which also inspires to try to factor of from obtaining



Therefore,

(8) Example (Finding Limits Using Trigonometry) Compute the limit of at

    Solution. We have

(9) Example (Finding Limits By Rationalizing) Compute:
    
    Solution. We have
    

(10) Example (Finding Limits By Rationalizing) Compute the limit of at

    Solution. We have

(11) Example (Finding Limits By Rationalizing) Compute   

    Solution. We have
    
    
    
    
    
    
    
    
    
    
    
    

(12) Example (Finding Limits By Rationalizing) Compute the limit of at

    Solution. We have

(13) Example (Finding Limits of Piecewise Functions) Find the limit,



    Solution. Since we know that and so we use to evaluate the limit of the piecewise function, as follows,
    

(14) Example (Finding Limits of Piecewise Functions) At compute the limit of



    Solution. Since the function is pieced together at we will evaluate two one0sided limit. First the limit from the left, we have and for the limit from the right we have Since
    


we know the two-sided limit must exist and we have even though

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