The limit of a function
The definition of the limit of a function
A limit on the left (a left-hand limit) and a limit on the right (a right-hand limit)
Continuous function
Limits at infinity (or limits of functions as x approaches positive or negative infinity)
Infinite limits
The limit of a function examples
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 the point 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
Continuous function
A real function  y = f (x)  is continuous at a point a if it is defined at x = a and
that is, if for every e > 0 there is a  d(e) > 0 such that  | f (x) - f (a) | < e  whenever  | x - a | < d(e).
Therefore, if a function changes gradually as independent variable changes, so that at every value a, of the independent variable, the difference between f (x) and f (a) approaches zero as x approaches a.
A function is said to be continuous if it is continuous at all points.
Limits at infinity (or limits of functions as x approaches positive or negative infinity)
We say that the limit of  f (x) as x approaches positive infinity is L and write,
if for any e > 0 there exists N > 0 such that  | f (x) - L | < e  for all  x > N(e).
We say that the limit of  f (x) as x approaches negative infinity is L and write,
if for any e > 0 there exists N > 0 such that  | f (x) - L | < e  for all  x < -N(e).
Not all functions have real limits as x tends to plus or minus infinity.
Thus for example, if  f (x) tends to infinity as x tends to infinity we write
if for every number N > 0 there is a number M > 0 such that  f (x) > N  whenever  x > M(N).
Infinite limits
We write
if  f (x) can be made arbitrarily large by choosing x sufficiently close but not equal to a.
We write
if  f (x) can be made arbitrarily large negative by choosing x sufficiently close but not equal to a.
The limit of a function examples
 Example:  Evaluate the following limits; Solution:  a)  As x tends to minus infinity f (x) gets closer and closer to 0. As x tends to plus infinity f (x) gets closer and closer to 0. Therefore, b)  As x tends to 0 from the left  f (x) gets larger in negative sense.
As x tends to 0 from the right  f (x) gets larger in positive sense. Therefore,
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