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## Calculus 1

### Unit 1: Lesson 3

Estimating limits from tables

# Estimating limits from tables

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.C (LO)
,
LIM‑1.C.5 (EK)
When we're provided with an appropriate table of values of a function, we can use it to estimate the function's limit at a certain point.

## Want to join the conversation?

• Can someone explain why 3.68 is a better estimate than 4 for the limit?
• Because 3.68 is more accurate than 4.
3.68 probably isn't the exact limit either, but given the table it is our best bet.
• I thought at the limit there would be no value, it would be undefinied. So when the limit goes to 5, and g(x) is 6.37 in this problem, doesn't that mean the limit doesn't exist?
• The question reads: What is a reasonable estimate for the limit of 𝑔(𝑥) as 𝑥 approaches 5?

So, loosely speaking, we are looking for some value 𝐿 (the limit), such that when 𝑥 gets very, very close to 5 (without actually being equal to 5), then 𝑔(𝑥) gets very, very close to 𝐿.

Looking at the table we see that as 𝑥 goes from 4 to 4.999, 𝑔(𝑥) increases and gets closer and closer to 3.68 .
And, as 𝑥 goes from 6 to 5.001, 𝑔(𝑥) decreases and gets closer and closer to 3.68 .

Thereby it is reasonable to assume that the limit of 𝑔(𝑥) as 𝑥 approaches 5 does exist, and that it's value is 3.68 .
• Well in n the graph plotted in most of the questions,test,examples has two different points marked 1) A closed (solid) dot means the endpoint is included in the curve and 2)an open dot means it isn't ...My doubt is if it is the open dot then how does the answer here turn out to 3.68
• Since in question, it is asked "What g(x) approaching as x approaches 5?" When we check for limit from left hand side (i.e., values less than 5) for x = 4.999 (which is very close to 5) we get g(x) = 3.68. Similarly, if we check for limit from right hand side (values that greater than 5), for x = 5.001 we get g(x) = 3.68. So, limit from RHS and LHS are equal, therefore limit for g(x) as x approaches 5 is 3.68.

It does matter if g(x) is different at x = 5 or in your words there is open dot (or for sake of discussion, lets say undefined) because in question we not asked for value of g(x) at x=5 but what we g(x) is approaching as x approaches 5.
• It may have saved a lot of confusion if it had been stated in the beginning that the function was not continuous over the interval [4,6].
• Do we disregard the solid point on the graph?
• when using limits, yes, but when using the value of the function yes we use this as the y value
• How about the fact that both points are approaching the same value? Shouldn't that make it automatically undefined?
• If the two directions approach the same value then it is explicitly NOT undefined, it is that value. The one exception is when both directions approach infinity, because infinity is not a value.

You want both directions to approach the same value though
(1 vote)
• what about a graph like x=5, that is a straight line parallel to y axis. what's the limit as x approaches 5?
(1 vote)
• x=5 isn't a function of x, so asking about the limit as x goes to 5 is meaningless.
• Are there known precedures , defined to try and build an equation which has been formed using a data table for a f(n) ??
• There are infinitely many functions that pass through any given set of points, so no.

However, if you're given a set of n-many points, no two with the same x-coordinate, then there is a unique n+1-degree polynomial that passes through all of them.
(1 vote)
• Why we are adding the word "approximately" to the limit when we find the value of that? like "the approximate value of the limit of this function is 3.68".

So these values are not accurate/exact values of that limit?
If yes, how?
(1 vote)
• The issue is that you only have 4.999, but the limit can get closer to that, what would 4.9999 or 4.99999 be? The assumption from the given values is that there will not be a big jump past 3.68, but there might be a small increment such as 3.681 that would make it only an approximate value. Hope this makes sense.