Limits & Continuity
- Ken Schwartz
Graphical illustration of limits and continuity. Moving slider or running animation makes approach from both sides. 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 blue dot to the value of "" where the limit and continuity are to be evaluated. As you move the slider (or run the animation), two values of will approach : one from the left (negative side), and one from the right (positive side). As gets closer to , we observe what happens to the corresponding values of these two points (the "one-sided limits"). If the -values appear to be "converging" together (to the same value), as gets close to , then we say the limit exists. If one or the other one-sided limits does not exist, or they do not both "converge" to the same -value, then the limit does not exist. This is indicated by a blank value in the corresponding expression in the right pane. A function is continuous at a point 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 (vertical asymptote), or if there is no -value at the point . More formally, is continuous at only if both one-sided limits exist at , those limits are equal to each other, and they are equal to the function value () at the point. 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 -value to another at integer values of . Set the blue dot to an integer value of and notice that the two one-sided limits converge to different -values as approaches . Thus, the limit does not exists at integer values of , and is not continuous at these values of . However, move the blue dot anywhere between integer values of , and the two one-sided limits both converge to the function value , so is continuous at these values of . The last function in the list has a "hole" at . Even though the hole is not shown on the graph, you will see the small blue dot on the graph disappear at . But notice that the limit exists at . The function is not continuous there, however, because does not exist (thus the hole). The other three functions are continuous everywhere, and so the limit must exist for all values of .