Limit of a function: Pinching theorem with streamlined method of solution

I discuss and solve an example involving the limit of a function and an application of the pinching theorem. Such a theorem involves inequalities and is seen in 1st year university mathematics. Note that I present a streamlined method of solution. As a result there is no need to split the method of solution into the cases: $x$ is positive; and $x$ is negative.

Category:

Education

Tags:

• pinching
• theorem
• squeeze
• sandwich
• calculus
• mathematics
• Chris
• Tisdell
• MATH1131
• UNSW
• Sydney

[avatarhzh527]

@DrChrisTisdell. Sorry I can't see why it isn't true. Since -1 <= sin(1/x) <= 1 wouldn't this mean that the inequality holds. There's probably something I'm missing. Can you please explain?

[DrChrisTisdell]

@avatarhzh527 My advice is to carefully listen to what I say at 01:12 and have a think about it.

[avatarhzh527]

Can't you just multiply by x anyway since you're taking the limits to zero and it doesn't really matter how the inequalities end up since you're using the squeezing theorem?

[DrChrisTisdell]

@avatarhzh527 Sorry, I'm not sure if I understand your question. If you mean that multiplying by x then you obtain

-x \le x sin (1/x) \le x for all x

then this isn't true - can you see why?

[crazyop2]

Thanks for the video. It showed the proof in a concise and logical way.

[DrChrisTisdell]

You're very welcome. Thanks for commenting. Glad that you appreciate it!

This method of proof isn't for everyone because of the unified inequalities.

[IronHuge]

This is great! Thank you. I was recently wondering what proof really is, after a sleep and watching this video, suddenly I realize that a proof is not only a deduction process that leads to a result, it also involves with an image already built in the mind. In my opinion, a proof could look tricky, but what is important is the mental image before the literal proof.

[scottsanerd]

Excellent. Clear, concise, easy to understand and follow.

[mroxen90]

thats fantastic...howeva...hoho...ur explanation mayb still confused me a bit...

but, it really help me a lot to understand why is it like that...thankz....err...may i ask u a question, sir... is pinching theorem and squeeze theorem is the same?