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# One-sided limits from tables

Sal finds the one-sided limit of x²/(1-cosx) as x approaches zero from the right, using a table of values. Created by Sal Khan.

Video transcript

Consider the table
with function values for f of x is equal to x
squared over 1 minus cosine x at positive x-values near 0. Notice that there is one
missing value in the table. This is the missing
one right here. Use a calculator to evaluate
f of x at x equals 0.1 and enter this number in the
table rounded to the nearest thousandth. From the table, what does the
one-sided limit, the limit as x approaches 0 from the
positive direction of x squared over 1 minus cosine
of x, appear to be? So let's see what they did. They evaluated when x
equals 1, f of x is 2.175. When x gets even a
little bit closer to 0-- and once again we're approaching
0 from values larger than 0, that's what this little
superscript positive tells us-- we're at 0.5 and we're at 2.042. Then when we even closer
to 0, 0.2 f of x is 2.007. I'm guessing when I'm
getting even closer it's going to be even
closer to 2 right over here. But let's verify that. Get my calculator out. So I want to evaluate
x squared over 1 minus cosine of x when
x is equal to 0.1. Let me actually verify
that I'm in radian mode, because otherwise, I might
get a strange answer. So I am in radian mode. Let me evaluate it. So I'm going to have 0.1
squared divided by 1 minus cosine of 0.1, and
this gets me 2.0016. And let's see, they
want us to round to the nearest thousandth. So that would be 2.002. Type that in, 2.002. And so it looks like the
limit is approaching 2. It's not approaching
2.005, we just crossed 2.005 from
2.007 to 2.002. So let's check our answer,
and we got it right. I always find it fun to
visualize these things. And that's what a graphing
calculator is good for. It can actually graph things. So let's graph this
right over here. So go in to graph mode. Let me redefine
my function here. So let's see, it's
going to be x squared divided by 1 minus cosine of x. And then let me make sure that
the range of my graph is right. So I'm zoomed in at the right
the part that I care about. So let me go to the range. And let's see, I care about
approaching x from the-- or approaching 0 from
the positive direction, but as long as I see values
around 0, I should be fine. But I could actually zoom
in a little bit more. So I could make my minimum
x-value negative 1. Let me make my maximum x-value--
the maximum x-value here is 1, but just to get some space
here I'll make this 1.5. So the x-scale is 1. y minimum, it seems like
we're approaching 2. So the y max can
be much smaller. Let's see, let me make y max 3. And now let's graph this thing. So let's see what it's doing. And actually it looks --
whether you're approaching from the positive direction or
from the negative direction-- it looks like the value of
the function approaches 2. But this problem, we're
only caring about-- as we have x-values
that are approaching 0 from values larger than 0. So this is the one-sided
limit that we care about. But the 2 shows up right
over here, as well.