Piecewise Functions and Limits
Now we'll shake things up a bit by taking limits with piecewise-defined functions. Here's an example:
What is ?
If we draw the graph of this function, we see that it looks like the line y = x + 1 except at one point. When x = 1, instead of having y = 2 like we would expect, the point has jumped off the line up to y = 3.
How does a function like this affect what we know about limits? Imagine we're taking Bruno, the ugliest dog, for a walk. We would expect him to stay on the sidewalk. We wouldn't expect him to suddenly teleport to Middle-earth, then reappear and continue on his path. He may look like Gollum, but still...
When talking about limits, we're talking about what we expect the function to be doing. We assume Bruno is approaching solid ground.
In the example above, the limit is 2, because that's what we would expect the value of the function to be if we looked at values of x close to (but not equal to) 1.
We can think of as the value that f(x) gets "close" to as x gets close to 1.