It is good to use the concept of "approach" to explain limits of functions.
Let's evaluate the limit(x^2, x = 4).
1. Numerical approach
Evaluate the function f(x)=x^2 at several points near x=4 and use the results to estimate
the limit(x^2, x = 4).
| x | -3.9 | -3.99 | -3.999 | 4 | 4.001 | 4.01 | 4.1 |
| f(x) | 15.21 | 15.9201 | 15.992001 | 16 | 16.008001 | 16.0801 | 16.81 |
f(x) approaches 16. f(x) approaches 16.
From the results shown in the table, you can estimate the limit to be 16.
2. Graphical approach
The limit of numerical approach is reinforced by the graph of f (see Figure 01 below).
3. Analytically approach
The limit of analytically approach is using algebra or calculus.
There is the theorem that evaluate the limit(f(x), x=c), and it makes the limit(f(x), x=c)
be same as f(c).
From the theorem, the limit(x^2, x = 4) is 4^2, 16.
There are some limits that fail to exist.
If the function has behavior that differs from the right and from the left on the point,
the limit of the function on that point will not exist (See Figures 02 below).
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