Limits from tables
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Limits from tables
Consider the table with function values for f of x is equal to 3 sine of 5x over sine 2x at x values near 0. From the table, what does the limit as x approaches 0 of 3 sine of 5x over sine of 2x appear to be? So what they have is they're approaching 0 from below here. So first negative 0.1, and f of x is 7.239550. Then we get even a little bit closer to 0. x is negative 0.01. Once again, we're approaching 0 from values less than 0. And now we've gone up to 7.497375. We get even a little bit closer to 0, and we get to 7.499974. Now let's see what happens as we approach 0 from above, from values greater than 0. When x is 0.1, it's 7.239550, actually the same as this value right over here. When it is 0.01, it's 7.497375, actually the same value as here. And once again, when we're at 0.001, so we're approaching from above, from values greater than 0, 7.499974. So this is an interesting thing. The limit in both cases seems to be approaching, getting closer and closer to 7.5. And so you could imagine that this would be-- well, let's actually graph that just to verify it for ourselves. So we could type in our answer here and check our answer, and we got it right. But just to visualize what that looks like, let's actually get our graphing calculator. So I have the graph, just to show you how I get here. So if I were to start on the home screen, just press Graph, define my function of x. And we could just write 3 sine of 5x divided by sine of 2x. And then let me define my range. So, let's see. x, I want it to be-- yeah, sure, I could make it around 0. And let's see, I'm getting y values that go up to-- let's see, around 7 or 8. So I could make my y, let me do a little bit of negative y values. So let me start at negative 1 for my y range, and go all the way up to, let's say 8. Because this doesn't look like it's getting above 8, or at least in the range that we're talking about. And now let's graph this thing. So let's see what it does. OK, so the graph is doing all sorts of neat things, but then as we get closer and closer to 0, as x gets closer and closer to 0 from the negative direction, we see that the function is approaching 7.5. Likewise, when x is getting closer and closer to 0 from the positive direction, the function seems to be approaching 7.5. So you have this-- well, the function isn't defined at 7.5, or it's pretty clear that when you approach from both sides, you're getting there.