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# Limits from tables

Video transcript

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.