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## AP®︎ Calculus BC (2017 edition)

### Course: AP®︎ Calculus BC (2017 edition)>Unit 1

Lesson 2: Analyzing limits graphically

# Approximating limits

Before we learn about advanced methods for finding limits exactly, let's see how we can approximate a limit using graphs or tables.

## Want to join the conversation?

• Can't you just do the following?

(x^3 - 2x^2) / 3x - 6 =
[x^2(x-2)] / [3(x-2)] =
x^2 / 3
Substitute x with 2, and you get 4/3, which is the limit, though not the y value.
• I think it works, since the function is continuous and cancelling expressions in the numerator and denominator are conventional for other types of limit problems.

If you were to have lim x->5 (x^2-25) / (x-5), you can expand this to:

lim x->5 (x-5)(x+5) / (x-5) and the (x-5)s cancel. (The limit is 10). So if you can do it here, why not up there, if the function is continuous?

[For this particular example, you can also just use L'Hospital's Rule.]
• Since Sal got 4/3 as his answer, does that make 0/0 = 4/3? Also, can't you just factor an (x - 2) from the numerator and denominator to get x^2 / 3?
(1 vote)
• It is important to remember that a limit is not the same thing as the value of the function at that point. So no, 0/0 does not equal 4/3. The function just gets closer and closer to 4/3 as x gets closer to 2. However, when x is exactly 2, the function does not equal 4/3. It is undefined because we are not allowed to divide by 0.

Your strategy of factoring out (x-2) is exactly how we would figure out the limit algebraically. Again, we can only do this because we are looking for the limit, not for the function value itself. That's because we're not allowed to divide by 0, which is exactly what we would be doing to the numerator and denominator if we canceled out the (x-2).
• How would I solve:
lim (2x+1)/(3x-4)
x--> -infinity
Thank you!
• 1. Divide the numerator and denominator by x. As x--> inf, 1/x -->0, 4/x-->0 so the limit is 2/3.
2. Use L'Hopital's rule and differentiate the numerator and denominator w.r.t x ---> 2/3
• Is there any book you recommend?
• In my opinion, there isn't any book specifically recommended for the course except the course books that schools or colleges will require. Otherwise, we can find many exercises on google about calculus limits. I would rather stick to the college/school acquired book.
(1 vote)
• If the function is defined at the value, would you need to double check it with approximation? Or could you just stop there.

So in this case if when we plugged in x = 2 and it did not give us 0/0, it gave us lets say a 4. Would that be the limit? and it wouldn't need to be an approximation then?
• If the function is continuous, and the two-sided limit is the same as x approaches the value from both sides, then you're done, you've found the limit.
• how to find lim 7 when x approaches to 100
(1 vote)
• I understand that we are approximating the limits through graphs and tables . But what if we have to not use graphs and tables . Can't we do it in the way like lim x->2- f (x)
And lim x->2+ f (x)
Also if we do it we have to put x=2-h and x=2+h but ultimately answer turns out 0/0
(1 vote)
• In this case we have 𝑓(𝑥) = (𝑥³ − 2𝑥²)∕(3𝑥 − 6) = 𝑥²(𝑥 − 2)∕(3(𝑥 − 2))

We have a factor (𝑥 − 2) in both the numerator and denominator, so for any 𝑥 ≠ 2 we can write 𝑓(𝑥) = 𝑥²∕3

Also, for any ℎ ≠ 0 we have 𝑓(2 + ℎ) = (2 + ℎ)²∕3 = (4 + 4ℎ + ℎ²)∕3

As ℎ → 0, the terms (4ℎ) and (ℎ²) become negligible and we write
𝑓(𝑥) → 4∕3 as 𝑥 → 2
(1 vote)
• In this case, since the graph is a reflection, he could have just looked where x=-2. When x=2, the graph would be approaching the same point (1.333)
(1 vote)
• At , define approximate
(1 vote)
• This is what Sal Khan has made this video on. Approximating the limits basically means
"The generalization of the ordinary limit for real-valued functions of several real variables."
So he is trying to find the closest possible value to the destined limit. I hope this helps ya
(1 vote)
• So I went to Desmos and was able to pull up the same graph, but when I found the (2, undetermined) I could not make it stay. Does anyone know how?
(1 vote)