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
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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