Graphical illustration of limits and continuity.[br]Moving slider or running animation makes [math]x[/math] approach [math]c[/math] from both sides.[br]Right-hand pane shows existence and values of limits and function.

Select one of the functions from the drop-down menu in the right pane. Move the [color=#1551b5][b]blue[/b][/color] dot to the value of "[color=#1551b5][math]c[/math][/color]" where the limit and continuity are to be evaluated. As you move the slider (or run the animation), two values of [math]x[/math] will approach [color=#1551b5][math]c[/math][/color]: one from the [color=#b20ea8][b]left [/b][/color](negative side), and one from the [color=#0a971e][b]right[/b][/color] (positive side). As [math]x[/math] gets closer to [color=#1551b5][math]c[/math][/color], we observe what happens to the corresponding [math]y[/math] values of these two points (the "[b]one-sided limits[/b]"). If the [math]y[/math]-values appear to be "converging" together (to the [color=#c51414][b]same [math]y[/math] value[/b][/color]), as [math]x[/math] gets close to [color=#1551b5][math]c[/math][/color], then we say the [b]limit exists[/b]. If one or the other one-sided limits does not exist, [b]or[/b] they do not both "converge" to the same [math]y[/math]-value, then the [b]limit does not exist[/b]. This is indicated by a blank value in the corresponding expression in the right pane.[br][br]A function is [i]continuous[/i] at a point [math]x=[/math][color=#1551b5][b][math]c[/math][/b][/color] if its graph is "connected" at that point. A graph will not be connected if the two one-sided limits are not the same (indicating a break or jump in the graph), or if the graph at one or both sides of the point goes to [math]\pm \infty[/math] (vertical asymptote), or if there is no [math]y[/math]-value at the point [math]x=[/math][color=#1551b5][b][math]c[/math][/b][/color]. More formally, [math]f(x)[/math] is continuous at [math]x=[/math][color=#1551b5][b][math]c[/math][/b][/color] only if both one-sided limits exist at [math]x=[/math][color=#1551b5][b][math]c[/math][/b][/color], those limits are equal to each other, and they are equal to the function value [math]f[/math]([color=#1551b5][b][math]c[/math][/b][/color]) at the point.[br][br]Study different functions using the drop-down list in the right pane. The "floor" function creates a stair-step function, that instantly "jumps" vertically from one [math]y[/math]-value to another at integer values of [math]x[/math]. Set the [color=#1551b5][b]blue dot[/b][/color] to an integer value of [math]x[/math] and notice that the two one-sided limits converge to different [math]y[/math]-values as [math]x[/math] approaches [color=#1551b5][b][math]c[/math][/b][/color]. Thus, the limit does not exists at integer values of [math]x[/math], and [math]f[/math] is not continuous at these values of [color=#1551b5][b][math]c[/math][/b][/color]. However, move the [color=#1551b5][b]blue dot[/b][/color] anywhere [i]between[/i] integer values of [math]x[/math], and the two one-sided limits both converge to the function value [math]f([/math][color=#1551b5][b][math]c[/math][/b][/color][math])[/math], so [math]f[/math] is continuous at these values of [color=#1551b5][b][math]c[/math][/b][/color].[br][br]The last function in the list has a "hole" at [math]x=2[/math]. Even though the hole is not shown on the graph, you will see the small [color=#1551b5][b]blue dot[/b][/color] on the graph disappear at [math]x=2[/math]. But notice that the limit exists at [math]x=2[/math]. The function is not continuous there, however, because [math]f(2)[/math] does not exist (thus the hole).[br][br]The other three functions are continuous everywhere, and so the limit must exist for all values of [math]x[/math].