Intuitive Notion of the Limit

`f(x)` = `f(x)` =
In this applet, we see a function `f` graphed in the `xy`-plane. You can move the blue point on the `x`-axis and you can change `delta`, the "radius" of an interval centered about that point. The point has `x`-value `c`, and you can see the values of `c` and `f(c)`. You can type in your own functions in the left input box, or you can use the pre-loaded examples in the right drop down box.
We say `lim_(x-->c) f(x)` exists if all the values of `f(x)` are "really close" to some number whenever `x` is "really close" to `c`.


  1. Start by dragging the blue point on the `x`-axis. What is the relationship between the red segment on the `x`-axis and the green segment(s) on the `y`-axis?
  2. What does the `delta` slider do? Notice that `delta` does not ever take on the value of zero. You can "fine tune" `delta` by clicking on the slider button then using the left and right keyboard arrows.
  3. As `delta` shrinks to `0`, does the green area always get smaller? Does it ever get larger? Does the green area always shrink down to a single point?
  4. Try the various examples in the applet to get a good feeling for your answers in the previous problem.
  5. Example 5 shows a function that is not defined at `x = 1`. Even though `f(1)` has no value, we can make a good estimate of `lim_(x-->1) f(x)`. In this case, `lim_(x-->1) f(x)` tells us what `f(1)` "should" be. Use zooming to estimate this limit. (You can find instructions on panning and zooming here.)
  6. In Examples 6 and 7, the function is undefined at `x = 2`. (The function truly is undefined, even though the applet shows `f(2) = oo`. Check this yourself by plugging in `2` for `x` in the function). What is the value of `lim_(x-->2) f(x)`?
  7. Example 8 is a function that gets "infinitely wiggly" around `x = 1`. What happens if `c = 1` and you shrink `delta`? Try this: make `c = 2` and `delta = 0.001`. What will happen as you move `c` slowly toward `1`? Make a guess before you do it.

Project idea

Let `f(x)` be a function and define `g(x) = lim_(t-->x) f(t)`. Be careful to distinguish between `t` and `x`! You may have to read the definition of `g(x)` several times and think carefully about the situation. (This mixture of variables `x` and `t` comes up again later when we discuss integrals.)
  1. What is `g(c)` when `f` is continuous at `x = c`?
  2. What is `g(c)` when `f` has a removable discontinuity at `x = c`?
  3. What is `g(c)` when `f` has a jump discontinuity at `x = c`? Does it depend on whether or not `f(c)` is defined?
  4. What is `g(c)` when `f` has an infinite discontinuity at `x = c`?
  5. Give an example where the domain of `g(x)` is bigger than `f(x)`.
  6. Give an example where the domain of `g(x)` is smaller than `f(x)`.
  7. Give an example where `g` and `f` have the same domain.
  8. Is `g(x)` always a continuous function?
  9. Is it possible for `g(c)` and `f(c)` to be defined but not equal?