When the applet is first loaded you'll see functions `f` and `g`, and
the "Show limit control" box is not checked.
Check the box to show `f(x) + g(x)` and verify that the red function
being shown really is the sum of `f` and `g`. Drag the blue point on the `x`-axis,
and observe the correspondence between the numerical values on the
left and the graph on the right.
Now check the box "Show limit control" and slowly move `delta` to zero.
As `delta` goes to zero, the segments on the `y`-axis show us better and better
approximations for the limits of `f`, `g`, and `f + g` at the point `x = c`.
When the approximations are very good (when `delta` is very small), it is certainly
believable that the sum law for limits holds. Do you see how this is
reflected in the graph?
Uncheck the box to show `f(x) + g(x)` and check the box to show `f(x)g(x)`.
Again, verify that the red function really is the product of `f` and `g`,
and observe how the product law for limits is demonstrated when
`delta` is very small.
All of the above is reasonable at points where `f` and `g` are continuous
because `f(c)` actually is the limit of `f` as `x-->c`. But what about the
point `x = 3` where `f` and `g` are not defined?
Move the blue point so that `c = 3` and observe that none of the
functions are defined.
Even though the functions are not defined at `x = 3`, verify
that limit laws continue to be true at `x = 3`.