I Showed No Mercy And Crushed This Limit🥷
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- Опубліковано 19 чер 2024
- In this video, we'll use the squeeze theorem to crush this limit problem like a boss! Knowing how to solve limits of trigonometric functions on your calc ab exam is essential! My hope is that my videos supplement your precalc and ap calc learning so that you can take it to the next level with confidence! Remember the squeeze theorem sin(x) / x property as this useful identity can really simplify a lot of these problems. I hope you enjoy the video!
0:00 - Introduction
0:13 - Squeeze Theorem Formula
0:45 - Manipulating The Formula
2:25 - Solving The Limit Products
3:10 - U Substitution
before seeing this i was going to just use compound angle identity and waste 20 mins thanks for saving my time
Glad to help!
I used the fundamental theorem of engineering. sin(x) = x
So the limit is 5/4
Yup, that works! For very small x super close to 0, sin x approximates to x (also as seen from the Taylor series)
A other way to crush is is talyer expansion with limit to 0 means you can expand to the first term and rest rest is little o(x) then you have 5x/4x easy that is how i learned to solve almost all my limits by lineairlastion
Wouldn’t we first need to know that the limit exists?
No. When given a problem like this, we have to demonstrate that either it exists (and what it is), or prove it doesn’t. When you see a question like this on your AP exam, you may not know
What's the final answer you came up with?
I didnt understand how when you break up the fraction, 4x/sin4x ends up being sin4x/4x. How did the numerator and the denumerator swich places?
It was cross-grouping. It wasn’t so much flipping that single fraction as it was grouping terms almost like in an “x” across fractions
Based on when you guys are mostly online, I've moved my videos from Wednesdays to Saturdays! Hopefully this is better for you!
Why could one not use l'hopital's rule? Is LHR dependant on squeeze theorem to define it?
Exactly. If you use LHR here, you’re essentially using the definition of LHR here from squeeze theorem on itself so it’s more precise to not use LHR
@@tolkienfan1972 it’s incorrect to use the rule here because it’s circular logic using the definition on itself. Research it.
I’m aware of LHR and you can watch my other videos on it but it’s incorrect to use it here
@@tolkienfan1972 a candidate, yes, as it’s in indeterminate form, and for other function types to solve, indeed. But the presented limit here already is using a squeeze theorem method to solve the limit of the sin trigonometric function
By definition, LHR was derived using the squeeze theorem to observe the convergence of sin x / x for very small x
There really isn’t anything more to debate on this. Just like chat gpt hallucinations, we can’t simply quickly read something on Wikipedia or WA and run away with it as a confirmation bias. We need to look at the full problem at hand, and if you researched even further, you’ll see math forums discussing this
QED
@@tolkienfan1972 no it isn’t sufficient in this specific case of squeezing a trig function which is where lhr came from in the first place
QUIZ: Based on what I discussed in this video, for a non-zero value of k, what's the limit as x->0 of (sin(kx)) / kx
1
@@manav_chak7410 well done, ninja 🥷
Is the second time in my life (the first one was at University) that somebody "deduce" the result !??? Use l'Hôpital theorem, derivative in the numerator, derivative in the denominator. It's just 5/4 !!! We don't need to analyze anything
Watch my latest video published today and you’ll see why I didn’t use LH rule here.
@@NumberNinjaDave OK, I think you should have warned that we couldn't use that "magic" tool, the explanation has no sense.
@@DanielMartinez-ss5co thanks for the feedback, man 🙏