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Sequences and limits
 Operations with limits
  We usually use following results when finding the limit of a sequence.
Let {an} and {bn} be two sequences of real numbers such that an ® a  and  bn ® b. Then,
  Operations with limits examples
Let apply the above operations with limits to calculate limits of given sequences.
 Example:  Find
Solution:
 Example:  Find
 Solution:  
First factor the term of highest degree from both the numerator and denominator.
Note that the same procedure can be applied to every fraction for which the numerator and denominator are polynomials in n.
The limit of such a fraction is the same as the limit of the quotient of the terms of highest degree.
 Example:  Find
 Solution:  
 The limit of a function
 The definition of the limit of a function
The limit of a function is a real number L that f (x) approaches as x approaches a given real number a,
written
if for any e > 0 there is a  d (e) > 0 such that  | f (x) - L | < e  whenever  | x - a | < d (e).

The definition says, no matter how small a positive number e we take, we can find a positive number d such that, for an arbitrary chosen value of x from

the interval     a - d < x < a + d,
the corresponding function's values lie inside
            the interval     L - e < f (x) < L + e,
as shows the right figure.

That is, the function's values can be made arbitrarily close to the number L  by choosing x sufficiently close to a, but not equal to a.

Therefore, the number d, that measures the distance between a point x from the point a on the x-axis, depends on the number e that measures the distance between  f (x) from the point L on the y-axis.
 Example: Given
          whenever
A limit is used to examine the behavior of a function near a point but not at the point. The function need not even be defined at the point.
 A limit on the left (a left-hand limit) and a limit on the right (a right-hand limit)
The limit of a function where the variable x approaches the point a from the left or, where x is restricted to values less than a, is written
The limit of a function where the variable x approaches the point a from the right or, where x is restricted to values grater than a, is written
If a function has both a left-handed limit and a right-handed limit and they are equal, then it has a limit at the point. Thus, if
 
 
 
 
 
 
 
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