Let's look at a more complicated example...

Given this graph of f(x)...


First of all, let's look at what's happening around the dashed blue line. Recall that this is called a vertical asymptote.



So...     lim12a.gif

Another way to think about the limit is the find the height of the graph at (or really close to) the given x. Think about a little mountain climbing ant (call him Pierre) who is crawling on the graph. What is Pierre's altitude when he's climbing towards an x? That's the limit!

Let's try some more...


Let's look at what's happening at x = -7... The limit from the right is the same as the limit from the left... But there's a hole at x = -7!

That's ok! We don't care what happens right at the point, just in the neighborhood around that point. So...


Can you find the limit of f(x) as x approaches -3?

That's right!     lim14a.gif

How about the limit of f(x) as x approaches 0?

Right again!     lim15a.gif