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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 1

Lesson 4: Estimating limit values from tables

# Approximating limits using tables

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.C (LO)
,
LIM‑1.C.5 (EK)
We can have an estimate for a limit by evaluating the expression at points that are close to where the limit is, but there are a few things we have to consider before we do so.

## Want to join the conversation?

• Just to make sure, what exactly does the number 1.8 represent? It's not the asymptote so is it the Y value as it approaches 3? Does that mean that its going to be open when x=3 ?
• I'm new to this thing too, and because of this this isn't very articulate or eloquently written, but I think it is the value y is approaching as x approaches 3. Well, obviously, that seems quite obvious. But I have to specify this isn't the actual value, since it's actually undefined. In a graph, we represent it with a little circle to show that it isn't actually that number, whether it be undefined or some other number. So even though it's discontinuous, as x is approaching 3, y is approaching (in an intuitive sense) 1.8. It's not the value, but it's where we would graph that part of the line.
• I found a better way to do this to get an exact value. You can simplify the expression to by factoring the numerator and denominator, then canceling the common factor of (x-3) this gives x^2/5 plug in 3 and you get exactly 1.8! I understand the point of the video, but why didn't Sal mention this?
• Excellent point! Yes you are correct that the exact answer of 1.8 can be obtained algebraically. Sal does talk about this technique in a later lesson.
• So what exactly is the purpose of a limit?
• Basically, it allows you to use "infinity" in a sophisticated manner. Ever wondered how on earth we're able to calculate an infinite series in a lifetime?
• I see these comments are kinda new (23 days highest). How come these calculus videos are being made right now and not before?
• I believe it is KhanAcademy making newer videos to replace the older ones. For example, some of the starting videos in this section are quite old, so I think KA just reshot them to make them better and more current.
• I didn't quite get the idea of asymtote.
• An asymptote is when a line approaches a certain x or y value, but never quite reaches it.
It is hard to explain without having a visual to back me up, but if you have a graphing calculator, then plug in the equation "y = 1/(x^2)". The graph will show that as x approaches both positive infinity and negative infinity, the line approaches, but never touches, y = 0, which shows that an asymptote for this equation is y = 0. Also, as x approches 0, the line never quite reaches x = 0, showing another asymptote which is x = 0.
I would have given a video on the topic, but I couldn't seem to find one, sorry.
• At , I tried doing it on my calculator and it gives me some random answers. It giving me something like 5.99... Is the calculator supposed to be in some specific mode or am I just doing it wrong?
• Without seeing your calculator keystrokes, I don't know for sure what error you are making. Probably the most common error for this type of calculation is to enter the expression incorrectly, so that the calculator performs the operations in the wrong order. Calculator mode is not an issue here for this type of calculation (but degree mode vs. radian mode would have been an issue had the calculation involved a trigonometric function).

Make sure to include parentheses around the expression in the numerator and parentheses around the expression in the denominator, so that the calculator divides these entire expressions.

Also, you should test whether your calculator performs the standard order of operations. For example, if you type 5 + 2 * 3, your calculator performs standard order of operations if it displays 11, but does not do this if it displays 21. It is much preferable to use a calculator that performs standard order of operations.

Example on a TI-89 calculator that performs standard order of operations, for x = 2.999:
the keystrokes (2.999 ^ 3 - 3 * 2.999 ^ 2) / (5 * 2.999 - 15) yield 1.7988002, which is near 1.8.

If you are still getting answers far from the limit value of 1.8 on your calculator, check that you are using the same number of 9's (or same number of 0's) after the decimal point for each instance of x in the function. Also make sure you are including the decimal point for each instance of x in the function.

Have a blessed, wonderful day!
• At , wouldn't any number over three be a negative number? Since evaluating the expression with three gets zero, any number more than three would end in a negative fraction, like -1/-1, which is impossible. Please satisfy my grade 5 mind
(1 vote)
• I believe you are mistaken. Sal is saying that dividing by zero is undefined. While having a negative fraction might be unintuitive, it is very real and very possible. If you have two negatives in a fraction, it becomes a positive.
• This may be an oversimplification, but it seems to me that determining a limit in calculus if simply finding out when x is this, y is that, i.e., if x = 3, y = 8. Is that right, or is there more to it?
• What you're talking about is just plugging in function values. In a given case, the function value may or may not be the same as the limit.

A limit value is what the function output approaches as the x value approaches a certain number. If this number happens to be the same as the function value, we say the function is continuous at that point. But if the function has a hole or a jump at that point, the limit value will be different.