Intuitive Notion of the Limit
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`.
- 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?
- 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.
- 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?
- Try the various examples in the applet to get a good feeling for your answers
in the previous problem.
- 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
- 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)`?
- 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.
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.)
- What is `g(c)` when `f` is continuous at `x = c`?
- What is `g(c)` when `f` has a removable discontinuity at `x = c`?
- What is `g(c)` when `f` has a jump discontinuity at `x = c`?
Does it depend on whether or not `f(c)` is defined?
- What is `g(c)` when `f` has an infinite discontinuity at `x = c`?
- Give an example where the domain of `g(x)` is bigger than `f(x)`.
- Give an example where the domain of `g(x)` is smaller than `f(x)`.
- Give an example where `g` and `f` have the same domain.
- Is `g(x)` always a continuous function?
- Is it possible for `g(c)` and `f(c)` to be defined but not equal?