3 secret weapons for evaluating limits (limit as x approaches infinity & limits at x approaches 0)
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- Опубліковано 1 сер 2024
- I am going to show you 3 secret weapons for evaluating limits (with 13 examples, including how to find the limit as x approaches infinity and evaluate limit as x approaches 0)! These are very useful for you calculus 1 and calculus 2 class. They will make evaluating some limits so much easier. You do not have to use L'Hopital's Rule or do the limits by factoring/canceling. You can just do what I did in this video (especially if you are in my calculus class). Some math professors may these calculus limits tips and tricks, but I call them secret weapons. They are "The List", "The Fact" and "The Limit".
0:00 3 secret weapons for evaluating limits
0:10 The List: ln(x)≪x^p≪b^x≪x^x (for sequence: ln(n)≪n^p≪b^n≪n!≪n^n)
7:54 The Fact: the limit as x goes to infinity of (1+a/x)^(bx)=e^(ab)
16:42 The Limit: the limit as theta goes to 0 of sin(theta)/theta=1
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Thank you my greatest teacher I finally understand the limit, the sine angle with approaching to zero equal to one. This is a very good theorem solving problems. Thank you so much.
I'd love to see the geometric proof for "The Limit".
Same
Khan Academy has a video on it: ua-cam.com/video/5xitzTutKqM/v-deo.html
bprp also has one
It's actually one of the more interesting limits you'll come across, at least in high school level Calculus. In that, it's proof is not too technical, and extremely satisfying to prove it yourself
A neat way of writing "The List" rigorously, is just saying that ln(x)=o(x^p), and x^p = o(b^x) and b^x = o(x^x)
Please also define b to be greater than 1, or else b^x could be stuck at one or approach zero as x tends to infinity
@@Lightwar49 good point
You can use the L'Hopital's rule for "the limit" if you just define sine in terms of its series expansion. Obviously you'll then need to prove its relation to geometry but it's another story.
Keep making such vids👍
Gamma(x) lies between b^x and x^x
Including Gamma to the list could be useful for analysing complexity of algorithms
(in fact list of sequences is useful for analysing complexity of algorithms)
Another example for the limit
limit((1-cos(x))/x,x=0)
Hint use double angle for cosine and Pythagorean identity
I can’t help myself but for some of these limits my mind goes to algorithms. Like for your second example. Log x is a very good algorithm and quadratic time is a meh algorithm, so quadratic time take longer so it’s infinity.
Although O notation is usually taught using the algebraic definition, I like the limit definition better and you wind up with lots of limits from the “list.”
What do you mean algorithm?
@@anshumanagrawal346 an algorithm is a step-by-step process for solving a particular problem. Once we describe an algorithm for solving a particular problem, we analyze how much time the algorithm takes in terms of approximately how many operations the algorithm does in the worst case with respect to the input size. There are algorithms whose running time is like log n And there are other algorithms whose running time is like n^2. One way of figuring out which algorithm is going to be faster is to take the limit to infinity of the quotient of the two running times and if you get to zero it means that the top is a faster algorithm And if you get infinity it means that the bottom is a faster algorithm.
We can't use it, need to prove it every time by squeeze theorem🤣
No. Time to use Epsilon-Delta.
If you prove once every limit rule - addition, product, division with epsilon-delta, why the heck would you that again for concrete functions?😀
thanks for your videos, I don't take math in university but it's just so interesting to learn hahaha... I have a question though, when do we decide to use the list instead of L'H rule?
It simply depends which is easiest.
L'H is solid for 0/0 or ∞/∞ (if f' and g' are defined), but if you have to differentiate many times to stop it being indeterminate, or it's a pain to differentiate, then see if the list can be applied.
@@xinpingdonohoe3978 thanks for your reply! so am I right to say I should get the same answer using the list and L'H rule?
@@kelv_kei the list is will tell you which side of the fraction is bigger at limits to ∞, so it will easily tell you whether it'll be ∞ or 0. However, L'H can work with all limits, but only if it's properly indeterminate. For example, it wouldn't work with lim(x→∞) sin(x)/x, because that's 1/∞. However, it can figure out lim(x→0) x*cot(x) by putting the reciprocal of one on the denominator. This limit equals 1.
So, in essence, the list is good as seeing how functions will behave at ∞ (but will only give ∞ or 0), and L'H is good all over R (as long as it's indeterminate).
@@xinpingdonohoe3978 thank you for your explanation, it is very useful!!
@@kelv_kei no problem
Hey bro . There are some mic in market you can attach it to your close and use your both hand freely
Actually steve has said in one of his old videos that a reason for a big mic is that he felt awkward with both hands free cause he didnt know what to do sith the left one
And it's simply his look. Don't take it away from him.
I love you
so if it was one of the same over the same, say x^p / x^p, would you divide the coefficients of the top and the bottom
If the denominator power is bigger it is 0, if the numerator power is bigger it is inf
@yes oh i see what you mean. What if it was like a x^p expression where one is a sqrt, so to the power of 1/2, and the bases arent the same. Say 5x / √(16x^2 +4 ). Is there a certain rule for this?
@@bradfnx5317 You can simplify that to lim_x->∞ 5x/√(16x²), as the other terms become insignificant. So then you get lim 5x/4x = 5/4 as your answer.
Why does the +4 inside the square root become insignificant? Because, as x gets larger, that +4 changes the limit by less and less. This is generally true of all constant terms if another term is going to infinity.
The general rule is that you find the highest power of x in the numerator and the highest power in the denominator and can ignore the rest, as long as x->∞. I suspect brpr didn't bring this up because he already taught this elsewhere.
@@ZipplyZane Thank you. That makes it a lot easier to understand. I had test where it was that exact problem but it was lim x->-infinity of -5x / √(16x^2). the answer was positive 5/4 and i got it wrong. You cleared it up pretty well thank you sir
What topic of Calc 2 are you guys studying?
As of 9/30/21 I am in AP Calc AB and I'm planning to switch to BC, should I?
Improper integrals.
It depends! Have you learned all the calc 1 stuff already?
@@bprpcalculusbasics No, I have just started this year but I read on the College Board's website that the difference btwn the classes is just a matter of scope and not difficulty. What do you recommend?
By scope I mean how much content they cover
@@edu10th47 I think you will have to talk to your teacher if she/he lets you skip ap calc AB. You will be missing quite a lot from the AB stuff if you skip it. And why are you thinking about skipping?
bro chill there are 12 months in a year
😍
5
tnt 😂
@Devarsh Dey Math is fun.
Where did your hair go? Nooooooo! Did it converge to the origin?
I expected you to add some intuitive proofs as well :(
I'd like that