Explore

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 δ to zero. As δ 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 δ 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 δ 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.