Introducing Limits Informally Graph and Table
- Dr. Jack L. Jackson II
Limit of a Function as x Approches a Constant
Informal Definition of the Limit of a Function as x Approaches a Constant Given a function f and constants c and L we say that if and only if as the input value (x) gets closer and closer to c the output value (f(x)= y) gets closer and closer to L. In the App: Type in the formula of a function in the input box for f(x). The graph will appear. Enter a value for c in the input box or by adjusting its slider. The value of the function and the corresponding point (c, f(c)) will be graphed in red if it exists. Whether or not this point exists and its value are irrelevant in the consideration of the limit. A dotted vertical line will be graphed at x= c. We want to consider how the graph approaches this line. If the function has a limit then the graph will be approaching a single point on this vertical line (from both sides). Move the slider for delta letting it get closer and closer to 0. You will see two points graphed (c- delta, f(c - delta)) and (c + delta, f(c + delta)). As delta gets closer and closer to 0, i.e. the x-values of these two points get closer and closer to c, the two points will approach a single point (c, L) on the vertical line if and only if the limit of the function is L. Take a look at the values in the spreadsheet. This is basically just an (x, y) table for the function. However, x-values have been chosen to get closer and closer to c. Does it look like the y-values are getting closer and closer to some number, if so, that number is L. Try this out with several different values of c. It is possible that the limit could fail to exist for several reasons. It could have limit from the right and from the left, but these are different values. We call this a jump discontinuity. It could have a vertical asymptote from one or both sides. There are also some other possibilities of how a limit might not exist. If f(c) = limit of f as x approaches c then we say the function is continuous at the point. This requires that the function exists, the limit exists, and they are the same value at the point. If the function is continuous at a point, then the graph is one connected piece inside a small enough circle centered at (c, f(c)). If the function is not continuous at x = c, then it has a discontinuity at x = c. This will be some type of break, e.g. a hole in the graph, a vertical asymptote, or a jump discontinuity.