Informal definition of limits and one-sided limits and graphical examples.

Beginning with a graphical example of a function f(x)=(x^2-1)/(x-1), we introduce the concepts and notation for limits: left-sided limits, right sided limits and "ordinary limits". This function is a simple line with a hole in it, because we have to remove the problematic point x=1 from the domain. The left and right limits agree, and the ordinary limit exists.

Next we give the informal definition of limits and one-sided limits.

Finally, we give two graphical examples of limits for functions defined by graphs:

The first example is the unit step function, which is always zero when x is less than 0, then jumps to 1 for all non-negative values of x. The left sided limit of the unit step function is zero at x=0, the right sided limit at x=0 is 1, and because the left and right limits don't agree, the "ordinary" limit does not exist.

The second example is a piecewise defined function given graphically, and this one has two interesting points. At x=0 the graph has a hole in it, but the left and right limits at x=0 agree, so the ordinary limit exists. At x=2, the graph has a jump in it, so the left and right limits don't agree and therefore the ordinary limit does not exist.