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]. [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]. [br][br]In practice, we are typically concerned with determining whether a limit exists, and estimating its value, if it does exist. We do this by letting x approach c either [b]from the left[/b] ([math]x\to c^-[/math]) or [b]from the right[/b] ([math]x\to c^+[/math]) and observing the corresponding function values f(x). If the [b]one-sided limits[/b] [i]exist [/i]and [i]are equal to the same number L[/i], then the two-sided limit also exists and is equal to L. [br][br][math]\lim_{x\to c}f(x)=L\text{ if and only if }\lim_{x\to c^-}f(x)=L\text{ and }\lim_{x\to c^+}f(x)=L[/math][br]

Use the input boxes to enter formulas for the left (red) and right (blue) pieces of a piecewise defined function. (You can use the same function formula in both input boxes to get a "normal" graph.)[br][list][*]Use the input box for c to change the point where you will investigate the limits. [/*][*]Use the input box for x to set a starting point for your investigation. [/*][*]Use the x \to c button to have x move closer to c.[/*][*]Use the Trace On / Off buttons to leave a trace of the function values on the y-axis to observe whether they approach a particular value. Use Clear Trace to remove all trace points. [/*][*]When the function values get "close enough" to the limit value, the one-sided limit notation will be displayed on the bottom-left of the screen. [/*][/list]