This lecture will introduce a new concept, the limit. A limit, essentially, is what a function "should be" for a certain input. Let's look at an example. Consider the function
If we factor the quadratic in the numerator, we see that and hence for almost all values of x, the function is simply .
But at the crucial point , the denominator and so requires division by zero and hence is undefined. However, since the function satisfies for , it isn't hard to guess by the above informal description of a limit that the limit of f at x=1/2 is . We write: