Use the input boxes on the left to define a function by entering the numerator and denominator functions. Note the values excluded from the domain of the function. The checkbox will show/hide the graph of the function. [br][list][*]Use the slider tool or input box to move the point x near to the excluded values. Notice the point leaves a trace of the graph of the function. (Click "Clear Trace" button to erase it.) [/*][*]Observe the behavior of the function near the excluded values. Note: the function values are displayed in function notation under the input box for x. [/*][/list]

Although almost all of Calculus relies on the limit concept (from a logical, theoretical standpoint), it was actually the last major concept of Calculus to be developed formally. This means that [i]for around 200 years (mid 1600s - mid 1800s) people were using Calculus without a formal approach to the limit concept[/i]. This is why I feel justified in choosing to focus on the informal, intuitive conception of limits. [br][br]We know that functions often have "domain issues" (e.g., an input that doesn't produce an output). For example, rational functions are undefined where the denominator is equal to 0 because division by 0 does not produce a numerical value for the output. When we find such excluded values (i.e., values not in the domain of the function), a natural question to ask is: "What does the function do (i.e., how does it behave) when the input is [i][b]near [/b][/i]the excluded value?" [br][br]An informal definition: We say that a function [math]f[/math] [b]has a limit[/b] [math]L[/math] as [math]x[/math] [i]approaches [/i][math]a[/math] if we observe [math]f(x)[/math] to become progressively closer to [math]L[/math] whenever [math]x[/math] gets progressively closer to [math]a[/math]. Notice that the limit is a numerical value (not a process). The notation for this is: [math]\lim_{x\to a}f(x)=L[/math].