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AP®︎/College Calculus BC

Course: AP®︎/College Calculus BC>Unit 4

Lesson 7: Using L’Hôpital’s rule for finding limits of indeterminate forms

L'Hôpital's rule: limit at infinity example

AP.CALC:
LIM‑4 (EU)
,
LIM‑4.A (LO)
,
LIM‑4.A.1 (EK)
,
LIM‑4.A.2 (EK)
Sal uses L'Hôpital's rule to find the limit at infinity of (4x²-5x)/(1-3x²). Created by Sal Khan.

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• This is just more of a general question, but why doesn't infinity divided by infinity just equal 1?
• Well, one reason is that two quantities could both approach infinity, but not at the same rate. For example imagine the limit of (n+1)/n^2 as n approaches infinity. Both the numerator and the denominator approach infinity, but the denominator approaches infinity much faster than the numerator. So take a very large n, like 1 trillion. The numerator is 1,000,000,000,001. But the denominator is 1 trillion SQUARED. So (n+1)/n^2 for n=1 trillion is .000000000001000000000001. That's very close to zero, not 1. As n gets even bigger, the limit of (n+1)/n^2 approaches even closer to 0.
• at when Sal says to evaluate for infinity, does it matter if it comes out as infinity/infinity or pos. infinity/ neg. infinity in order for l'hopitals rule to still apply?
• L'Hopital's rule can be applied in all cases of indeterminate forms which include 0/0, infinity / infinity, (infinity) /(-infinity) and
(-infinity) / (infinity).
• Sal my teacher told me a short cut. If f(x) is a approaching a infinity and the powers are the same (ex. 4x^2/3x^2)then the coefficients become what the limit approaches. Without even factoring you can tell that the answer is going to be 4/3. Does that work all the time?

Also thank you. I learned someone awesome in this video. :) As usual.
• The first time he introduced limits approaching infinity, that was the way he did it.
• How do we know, when are we supposed to use L'hopital's rule????
and when just normal limit??

How many times do i have to make :'hopital's rule, to get the result?? (in this harder cases?) Coz i can just make again and again, and what if i dont get the result?
• L'hopital's rule is just one of many different methods to get the result for a limit. If you see you are not getting results with it, try another approach. There is no way to tell the best way to solve an limit, that's just trial and error.
• Sal used L'Hopital's Rule twice. I solved the question by using L'Hopital's Rule once.

When I apply the limit (as x approaches infinity) to the derivative (8x-5) / (-6x) at , I know that "-5" is negligible when x reaches infinitely large values. That leaves (8x) / (-6x), which is scaling at the rate of 8 / -6, or 4 / -3. Hence, the answer to the question asked.

Can somebody please confirm if my approach is correct? Thank you!
• It is correct. You did what Sal said at the begining of the vidieo about already knowing how to get the limit. While your way is correct, Sal only continued taking L'Hopital's rule one step further to get the same result.
• at Sal says that "It doesn't matter what value this [limit] is approaching." I'm not sure I understand how that could be? The fraction would not be there at any point besides infinity, because otherwise it wouldn't be indeterminate, to my thinking. Or am I just reading too much into it? Thanks.
• At that point in the analysis, we're looking at the limit of a constant as x goes to infinity, and Sal is making the point that a constant will always have the same value no matter what limit we're approaching. He isn't saying that the specific limit (infinity) is irrelevant to the overall problem, but just noting that once you've got a constant you're done and don't have to worry about the limit any more.
• Hey!! I am learning sooooo much more from these presentations than I do in my lectures.... can we also us this rule to find the vertical and horizontal asymptotes? how do we use it from the + and - sides?
• I see the relationship you're foreseeing, since both deal with the slope of functions where values zero and infinity occur. For horizontal asymptotes, I suspect you may be right, that L'Hopital's Rule can help us find the value f(x) for which the slope approaches 0.

However, I don't believe the two are related when it comes to vertical asymptotes, since these deal with slopes of undefined forms (e.g., 5/0 or the general form n/0), not of indeterminate form (i.e., 0/0 or infinity / infinity).

Hope this helps!

Also, if you'd like a general review of horizontal and vertical asymptotes, you can watch the video below (although it's rather long, at 20 minutes, I'm sure you can parse through it).

http://tinyurl.com/qxwpcu7
• Hi. This question may be way off, but when we are calculating the derivative for (4x^2-5x)/(1-3x^2), why are we not using the quotient rule?
• You're applying L'Hopital's rule to calculate the limit at infinity. You're not interested in the derivative of this function (in which case, yes, the quotient rule can be used).
• i have a question how does infinity work infinity is well is infinity a number or is unlimited amount of everything