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 δ, 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.
Explore
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
limits exists?