The limit of a function is the value that the function approaches as [math]x[/math] approaches a value of [math]a[/math]. It is written as [br][math]L = \lim_{x\to\;a} f(x)[/math]. A limit can be a left limit or a right limit or both. A left limit is written as [math]\lim_{x\to\;a^-} f(x)[/math], a right limit is written as [math]\lim_{x\to\;a^+} f(x)[/math]. For a left limit the limit value, [math]a[/math] is approached from the left or '-' side and for a right limit the value is approached from the right.[br]This applet has several functions available by clicking the next button. Selecting "Left Limit" or "Right Limit" will approach from the left or right side, with neither checked the limit will approach from both sides. Clicking "Closer" will move closer to [math]a[/math]. The [math]\Delta[/math] value shown. With a one sided limit values of the function [math]\Delta[/math] and [math]\Delta/2[/math] from the limit are shown. If the value at the limit point exist it is also shown. In the theoretical world [math]x \ne a[/math].

Select a function then position [math]a[/math] where you want the limit. Values near boundaries are more interesting. Repeatedly click "Closer" to move [math]x[/math] closer to [math]a[/math]. Note the value of [math]f(x)[/math] shown in red. Then repeat from the other side ( Change "Right Limit" ).[br][br]Does the value approach a single constant value?[br][br]Is the value the same from both sides?[br][br]Is the value the same as the functions value [math]f(x=a)[/math]?[br][br]Note: If the left and right limit are the same the limit exist. If the limit exist and it is the same as the value of the function at the limit point, then the function is continuous at that point.[br][br]On a computer and calculators round off errors sometimes give incorrect answers. Can you find a situation where this applet gives wrong answers due to round off errors. This is especially true when subtracting nearly equal numbers or adding a very small number to a larger number.