If a function f(x) becomes arbitrarily close to a unique number L, the function trends toward L, getting increasingly close to it.
limit
Consider the following incomplete sentence.
If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, then the of f(x) as x approaches c is L.
In mathematics, we examine how functions behave as they approach specific values. The main goal is to observe whether they converge to a particular value L, grow without bound, or perhaps exhibit other patterns. Let's consider an example.
f(x) = x^2-3x
We are going to construct a table that shows the values of f(x) for two sets of x-values: one set that approaches c=3 from the left and one that approaches c from the right.
x
2.9
2.99
2.999
3
3.001
3.01
3.1
f(x)
- 0.29
- 0.0299
- 0.002999
0
0.003001
0.0301
0.31
We observe that as x approaches c=3 from either direction, the function f(x) trends toward L=0, increasingly getting close to it. Then we can say that f(x) becomes arbitrarily close to L as x approaches c from either side. Then, by the definition, 0 is the limit of f(x) as x approaches 3. Let's complete the sentence.
If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, then the limit of f(x) as x approaches c is L.
