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Functions, Limits, Continuity

Problems for "Limits"

Limits

Continuity

Problem :

f (x) =    


a) What is f (x) ?
b) What is f (x) ?
c) What is f (x) ?

a) f (x) = (x 3 -4) = 23 - 4 = 4 .
b) f (x) = (2x) = 2(2) = 4 .
c) Because f (x) = f (x) = 4 , f (x) exists and is equal to 4 .

Problem :

f (x) =    


a) What is f (x) ?
b) What is f (x) ?
c) What is f (x) ?

a) f (x) = (x 5 -12) = 25 - 12 = 20 .
b) f (x) = [(x + 1)3 -8] = (2 + 1)3 - 8 = 19 .
c) Because f (x)≠ f (x) , f (x) does not exist.

Problem : Evaluate:

   


Since we can find the limit of the function first, and then raise the limit to the power,


=      

Note that we can't just divide the limit of the numerator by the limit of the denominator, since the limit of the denominator here is zero. So, we must factor and reduce the rational function:


  =  
    = x 2  
    = 22  
    = 2  

Problem : Evaluate:

   


Here, the limit of the denominator goes to zero, so we can try to reduce the fraction:


  =  
    =  
    = = 4.  

Problem : Evaluate:

   


The limit of the denominator is zero. Here, we can try multiplying by the conjugate.


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