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Definition Of Limits

Limits Assignment / Homework Help
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.

Online Calculating Limits

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

Definition:  Symbolically, we write lim f (x) = L and read this as

"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 xa.

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

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