Limits

In this section we shall introduce limit from an institutive, non-formal point of view. The idea of limit is centered about the notion of “closeness”. We begin the development of the limit by the consideration of specific examples.

Example. Consider the function f defined by

f (x) = 2x + 1


Let us examine this function when x is near “3” but not equal to 3. Some values of f (x) for x less than 3 and then for x greater than 3 are given in.

x < 3                                             x > 3

f (2.7) = 6.4                             f (3.3) = 7.6 f ( 2.8) = 6.6                            f (3.2) = 7.4 f (2.9) = 6.8                             f (3.1) = 7.2 f (2.99) = 6.98                          f (3.01) = 7.02 f (2.999) = 6.998                       f (3.001) = 7.002

It is clear from and also from the graph of f in that as x gets closer and closer to 3, regardless of whether x approaches 3 from the left (x < 3) or from the right (x > 3), the corresponding values of f (x) get closer and closer to 7. We express this statement by saying that the limit of f (x) as x approaches (or, tends to) 3 is 7 and we write

 lim ( 2x + 1 ) = 7
x→3

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