The Limit Laws

In this applet we investigate two of the basic limit laws:
Sum Law:     `lim_(x-->c) (f(x) + g(x)) = lim_(x-->c) f(x) + lim_(x-->c) g(x)`

Product Law:     `lim_(x-->c) (f(x)g(x)) = (lim_(x-->c) f(x))(lim_(x-->c) g(x))`


  1. 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.
  2. 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`.
  3. 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?
  4. 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.
  5. 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`.