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]

Informally, [math]\lim_{x\to c}f(x)=L[/math] means that the function values f(x) get progressively closer to L the closer we let x get to c. 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]