Intuitive Notion of the Limit - One-Sided Limits
We see here the function `f` graphed in the `xy`-plane. You can move the blue
point on the `x`-axis and you can change `delta`, the length of an interval with one end
at 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 > c` and `x` is "really close" to `c`.
We say `lim_(x-->c-) f(x)` exists if all the values of `f(x)` are "really close" some number
whenever `x < c` and `x` is "really close" to `c`.
- Often, a one-sided limit exists even if a (two-sided) limit does not
exist. Which examples have points where this is the case?
- Can you think of a situation where a one-sided limit doesn't even exist?
Example 8 shows such a situation.
- Consider Example 9, a shifted square root. Does the left-hand limit exist at
`x = 1.5`? What about the right-hand limit? The normal (two-sided) limit?
- Is it possible for a limit to exist, but one of the one-sided limits does
not exist? Is it possible for a limit to exist, but neither of the one-sided