    # Definition Of Limits Now that we have looked at how limits arise when we calculate the area of a curved figure let us investigate the numerical and graphical methods of computing limits.

Consider the function f (x) = x 2 + x + 1

We will take a look at how f (x) behaves for values of near 2. The following table will give the values of f (x) for near 2 but not equal to 2. x f (x) 1 3 1.5 4.75 1.7 5.59 1.9 6.51 1.99 6.9501 1.999 6.995001

 x f (x) 3 13 2.5 9.75 2.2 8.04 2.1 7.51 2.01 7.0501 2.001 7.005001
From the graph and the table we can see that as approaches 2 from the left, f (x) gets closer to 7. Similarly as approaches 2 from the right f (x) approaches 7. We can further verify that f (x) can be made as close to 7 as required by taking sufficiently closer to 2.

In general we can say that
lim   x2+ x + 1 = 7
x->2

 Definition:  Symbolically, we write lim f (x) = L and read this as                                                        x->a "The limit of f (x) as x approaches a is L" If f (x) can be made arbitrarily close to L by taking x sufficiently close to a and x ≠ a.

We can also write this as f (x) -> L as x -> a.

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