The open circle says that the function doesn't have a value along the curve at x=1. And the closed, filled in circle is saying, "Hey, I'm the value of the function at x=1."
Big takeaway: It's possible for the function value to be different from the limit value.
And just because a function is undefined for some x-value doesn't mean there's no limit.
Holes in graphs happen with rational functions, which become undefined when their denominators are zero. Here's a classic example:
This is the graph of y = x / sin(x). Notice that there's a hole at x = 0 because the function is undefined there.
In this example, the limit appears to be 1 because that's what the y-values seem to be approaching as our x-values get closer and closer to 0. It doesn't matter that the function is undefined at x=0. The limit still exists.
Here's another problem for you to try:
What is a reasonable estimate for x→−4limf(x) ?
Reinforcing the key idea: The function value at x=−4 is irrelevant to finding the limit. All that matters is figuring out what the y-values are approaching as we get closer and closer to x=−4.
On the flip side, when the function is defined for some x-value, that doesn't mean that the limit necessarily exists.
Just like an earlier example, this graph shows the sort of thing that can happen when we're working with piecewise functions. Notice how we're not approaching the same y-value from both sides of x=3.