Introduction to Limits

Although the exact words and the letters used vary limits are usually defined in something like the following manner:

• There is a function f that is defined on an open interval that contains a point represented by c
• The function may or may not be defined at c (It is not required that f be defined at c)
• Then L is the limit of f(x) as x approaches c, written as:
• limx → c f(x) = L,
• if and only if for any δ > 0 there exists ε > 0 such that
• | f(x) - L | < δ when 0 < | x - c | < ε

What does this say? It says if we can pick an x near c that forces f(x) to get arbitrarily close to L then L is the limit of f(x) as x approaches c. In a later page we will present examples and provide problems that let you determine limits. Here we are going to pick values of x that get closer to c and see what happens to f(x). Note that x can approach c from below (x is less than c) or above (x is more than c). For the limit to exist f(x) must get closer to L as x approaches c from below and from above. If that is not the case then we do not have a limit as we have defined it here. Some examples are given below. You might first guess at the limit of the function then click on that function and the computer will give you the values of x and f(x) as x approaches some value.

 limx → 2 (x + 3) limx → 0 sin(x)/x limx → 3 (x2 - 9)/(x - 3) limx → 0 1/x limx → 0 [ 1- cos(x) ]/sin(x) limx → 3 (9 - x2/(x - 3)