Now that we know all of the cases for which a limit might not exist, let's go ahead and work through this example together. Feel free to try this on your own and then check back in with me. Now here in this problem, we're asked to use the graph to estimate the value of each limit or otherwise state that it does not exist.

Now the first limit that we're asked to find here is the limit of \( f(x) \) as \( x \) approaches negative 2. So looking at our graph here, my \( x \) value of negative 2 coming in from that left side, getting really, really close to negative 2, I see that my function seems to be approaching a value of 0. Then coming in from that right side again getting really, really close to negative 2, I see again that my function seems to be approaching a value of 0. So the limit of \( f(x) \) as \( x \) approaches negative 2 is equal to 0. Now, something that might trip you up here is that we have sort of a piecewise function going on. So once \( x \) is equal to negative 3, something completely different is happening. So you might be tempted to say, okay, well, as \( x \) is approaching negative 2, it's doing something crazy over here. So this limit does not exist. But you would be wrong in saying that because remember that when looking at limits, we are looking at values really, really close to whatever \( x \) value we're looking at. So negative 3 and negative 2, even though you might think of these values as being pretty close together, remember that we're looking at values really, really close to negative 2, like negative 2.01 or negative 2.001. So here, our limit is equal to 0. We don't have to worry about what else is going on with our graph over there. Now let's move on to our next limit.

Here, we're asked to find the limit of \( f(x) \) as \( x \) approaches 0. So on our graph here, as \( x \) approaches 0 from that left side, I see my function is approaching a value of 1. Then coming in from that right side, I see the same thing happening. As \( x \) gets really, really close to 0, my function is approaching a value of 1. So here, my limit of \( f(x) \) as \( x \) approaches 0 is 1. Now, something else that might have tripped you up here is that we have this hole in our graph. So the actual value of our function when \( x \) is equal to 0 is actually negative 2. But remember that this does not matter because when looking at our limit, we are looking at what our function is doing as \( x \) gets really, really close to 0, not when \( x \) is 0. So our function or our limit here is 1, and we can move on to our next limit.

Now the final limit that we're asked to find here is the limit of \( f(x) \) as \( x \) approaches negative 3. Now looking at our graph here, I can already see that something a little crazy is going on here around \( x \) equals negative 3, but let's dig a bit deeper here. Now looking at \( x \) approaching negative 3 from that left side, I see that my function is approaching a value of 1. Now from that right side, it is definitely not doing that. And I see that as we get really, really close to negative 3 coming in from the right, then my function is actually experiencing some unbounded behavior. And my function is going to negative infinity. Now remember that here when we're dealing with unbounded behavior, our limit does not exist. Now that's just our right sided limit. But we have another problem here because from the left and the right, our function's not doing the same thing. So we already know that our limit of \( f(x) \) as \( x \) approaches negative 3 simply does not exist because our one-sided limits are not the same. And one of our one-sided limits is actually already doesn't exist because it's experiencing unbounded behavior going to negative infinity.

Now let me know if you had any questions here, but let's continue on with some practice problems. I'll see you in the next video.