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