Cite This Page
To Go
Functions, Graphs, and Limits
Functions, Graphs, and Limits
group rates for schools and districts

Piecewise Functions and Limits

Now we will shake things up a bit. Here's a piecewise-defined function:

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 Chinese crested 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, 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.

Next Page: One-Sided Limits
Previous Page: Graphing and Visualizing Limits

Need help with College?