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### Course: Calculus, all content (2017 edition) > Unit 1

Lesson 2: Limits from tables# Limits from tables

Sal finds the limit of 3sin(5x)/sin(2x) at x=0 using a table of values. Created by Sal Khan.

## Want to join the conversation?

- Is there a pedagogical reason to be asking these kind of limit questions? Specifically the kind where we need to pick out what we feel a number is approaching without having a solid mathematical basis beyond guesstimation. It seems pretty arbitrary and potentially misleading to have the correct answer be something "nice" like 7.5 when there are infinitely many other numbers the limit could have been, like 7.4999999908, 7.5000001, etc.(20 votes)
- Probably the main reason for asking these type of limit questions is to introduce students to the intuitive idea of a limit, until the students have learned precise methods of finding limits. Guesstimation could also be used to check if a calculation of a limit (using a precise method) is likely to be correct, or if there's likely a mistake in the calculation.

Yes, I certainly agree that using mathematically precise methods of calculating limits is much preferable to finding limits by just using guesstimation.(19 votes)

- the graph drawn in the video shows a value( nearly 7.5) when X=0. But is it not undefined at x=0? then why does the graph show some value( rather an exact value) at x=0?(10 votes)
- Good question and you're absolutely right. However, my guess is that right now we're not talking about what the limit is, we're talking about what the limit "appears" to be instead - we're not using actual mathematics here, we're infrencing from the given numerical data.(6 votes)

- How can Mr Khan say that the function is approaching 7.5 and not 7.49? Is there a specific rule to round off your approximation to 2 sf?(2 votes)
- Note that for the values of x in the table that are closest to zero (0.01, 0.001, -0.01, -0.001), the function value is actually becoming farther from 7.49 and closer to 7.5 as x becomes closer to 0. So the limit is more likely to be 7.5 than 7.49 (though this does not prove for sure that the limit is 7.5).

This is still a good question, because a table can only suggest what the value of the limit might be, but cannot give a definitive value of the limit. This lack of certainty and precision from using tables is the reason why precise algebraic techniques and/or calculus techniques are used to find limits.

Have a blessed, wonderful day!(5 votes)

- What's the correct way of reading this? Is it 3(sin(5x))/sin(2x) or (3((sin(5))x))/((sin(2))x). It seems like the top was resolved as the first form while the bottom was resolved as the second form.(3 votes)
- Either way will result in the same answer from a calculator. Because the 3 is a factor in the numerator, you can think of multiplying sin(5x) by 3 and then dividing by sin(2x), or you can divide sin(5x) by sin(2x) and then multiply that result by 3. Both ways get you the same answer. Remember that division and multiplication are inverse operators of each other, and so in "order of operations" they are on equal footing -- you do them in the order you encounter them, moving from left to right.

That being said, you do have to be careful, as it can seem, at a first glance is if something different may be happening when your numerator has multiplication operators from when your denominator has multiplication operations. For a calculator, 12/2*3 will equal 18, and 12/(2*3) will equal 2. Here the parenthesis in the denominator change the order of operations, 2*3 is done before 12*something. But 3*4/2 will equal 6 and (3*4)/2 will also equal 6. In this case the 3*4 is always done first, the parenthesis don't matter, because 3*4 is encountered first when working from left to right.

To avoid any problem with this I always put the whole numerator in parenthesis, and the whole denominator in parenthesis -- so that what I want to happen, and what the calculator does, is always the same thing.(3 votes)

- How can we solve this question without using calculator?(2 votes)
- Without using a calculator, you can use l'Hopital's rule to derive the numerator and denominator to say the limit as x approach 0 of 3(sin(5x))/sin(2x) is the same as the limit as x approaches 0 of 3(5cos(5x))/(2sin(2x))=3(5cos(0))/(2cos(0))=3(5*1)/(2*1)=15/2=7.5(3 votes)

- How can we find the sin,tan without calculator?(2 votes)
- Check out this section of Khan: https://www.khanacademy.org/math/trigonometry/less-basic-trigonometry

particularly regarding the Pythagorean identity and "SOHCAHTOA", that is:

Sine = Opposite/Hypotenuse

Cosine = Adjacent/Hypotenuse

Tangent = Opposite over Adjacent

You should be quite solid in your trigonometry by now. Remember, "there is no learn it and forget it" in math.is in preparation for, and use in, something else. Trigonometric identities will come up in polar, cylindrical and spherical coordinate systems and are also very useful for solving integrals.**EVERYTHING**(2 votes)

- When he graphed the function (2:35), how come the function didn't show it is undefined at x=0? Does that mean that functions not only be naturally limited, but can also have a limit inserted on them?(2 votes)
- You're right it should be undefined at x=0. The graph doesn't show it probably because of limitations of the graphing calculator.

I'm not sure what you mean by "naturally limited". In fact, functions are not said to be limited, they have a limit at a certain input: the value they approach as you get closer and closer to that input (without actually reaching it). So here, the limit at x=0 is 7.5, regardless of what the actual value at x=0 is.

Note how I'm careful not to say anything specific about the actual value. In this case, the value is undefined at x=0, but it's possible to have a real value at the point where you're taking the limit. It's even possible that this real value is different from the limit! A simple example uses two equations:

f(x) = { y=(3sin(5x))/(sin(2x)) at x different from 0, y=0 at x equal to 0 }

This is a perfectly valid function, and the value at x=0 is 0, but the limit as x approaches 0 is 7.5! When graphing this, mathematicians usually put a visible dot at x=0, y=0 to show that the function has a so-called singularity there.(3 votes)

- I tried entering the calculation shown at0:22using x=-0.1, but the answer my calculator gave me as a result was 7.49992 rather than the 7.239550 that was shown in the video. Am I entering it into my calculator incorrectly or did I miss something?(3 votes)
- How can we tell if a limit exists by reading a table?(1 vote)
- You can see that as x is approaching 0 from the positive and negative direction, it is approaching a constant.(4 votes)

- Hi Sal.

How can I work this out without using a Calculator?(1 vote)- Without using a calculator, you can use l'Hopital's rule to derive the numerator and denominator to say the limit as x approach 0 of 3(sin(5x))/sin(2x) is the same as the limit as x approaches 0 of 3(5cos(5x))/(2sin(2x))=3(5cos(0))/(2cos(0))=3(5*1)/(2*1)=15/2=7.5(4 votes)

## 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.