Limits of Oscillating Functions and the Squeeze Theorem

Description:
Some functions start oscillating "infinitely" quickly near a point. Limits at those points don't exist if the oscillations have a nonzero height. However, of the function both oscillates and goes down towards zero, the Squeeze Theorem lets us compute the limit too.

Learning Objectives:
1) Compute the limit of a function near a point with "infinite" oscillations
2) Apply the squeeze theorem - carefully verifying the assumptions - to compute limits of functions such as xsin(1/x) near 0.

Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master calculus means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

This video is part of a Calculus course taught by Dr. Trefor Bazett at the University of Cincinnati.

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[maxwellchiu]

It is kind of amazing how everything make more sense after somebody like the Dr. presents it in a way that I've never seen before. Same stuff, just different, more illuminating.

[renewd]

What I love about the example of the limit of sin(1/x) is that it's almost as if the graph is oscillating before our very eyes and will never stop.

[motivator]

I have started understanding actually what is happening in calculus that too after graduating in maths. Accept my gratitude sir.

[mariogutierrez]

Great video! I understand the squeeze theorem in a more interesting way now!

[StephenHsiang]

what a clear and inspiring calculus course, thanks!

[Anne-gd]

Such a weird behavior of sin (1/×) is fascinating to me. .yeah! limits of oscillating functions and squeeze theorem is amazing thrill to do...
Thanking you sir for your valuable info about such spectacular phase of mathematics.

[loth]

Thanks a lot, this is exactly what I needed

[nathaniel]

Thanks, this video helped me finally understand the Squeeze theorem! I liked the combination of the visual and algebraic explanations.

[AdityarajXIC]

You know what you are just amazing Thanku sir love from India 🤙✌️✌️✌️✌️✌️

[awesomeswordsman]

thanks a lot sir, i've mostly breezed through highschool by memorizing a lot, and got really far by doing so, i was one of the top in my class, and now that i'm in college i regret it because at first i didn't understand anything at all, but thanks to kind teachers like you on Youtube, i was able to also get very far in college, with good grades and i sincerely thank you for helping me understand what i couldn't in highschool

[tusharnikose]

Amazing explaination for everything. Im kinda loving mathematics onwards. Thanks a lot . Hoping more videos like this or full detail course related to IIT-JEE syllabus . lm from India too. XD

[priynshu]

Very beautiful explanation.. Thank you sir

[proggenius]

I kind liked the squeeze theorem. Sounds interesting. Thanks Pr. Bazett.

[anupamkumar]

Excellent explanation sir love from india🇮🇳

[nathanielkilmer]

I came across the function sin(tan(x)) yesterday, and I became curious if there is a field of math that deals with such functions. I recalled learning about this particular function, sin(1/x) a long time ago when I was learning about limits, but I haven't seen any explorations of a function like sin(tan(x)). It might make a fun video, if you know enough to make a video on that topic (humble request).

[AustinPrince]

F(x)=Xsin(x) graph blew my mind, never knew that

[shivamrajpoot]

Nicely explained sir, I have understood everything very clearly... By the way, I am from India, lot of love from

[NOTsocreepyCREEPER]

i was taught this by the name sandwich theorem but i would say real good explanation

[mohamadxerullah]

Hello sir i have a question how will this thing come at an exam were the questions are all multiple choices

[iqbalmaruf]

Sir, i want permission for request. Proof for limit trigonometry for x approaches 0 is 1 like x/sin x & x/tan x. Please. I not understand for do it.