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Exercise two and this one states prove theorem two mentioned earlier in this video and we're given a hint use either definition two or

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This squeeze theorem here is a theorem two. I just copied and pasted. Let's just scroll up to see where this was found

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Where's the M2?

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Far away is it?

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So let's see how where it was

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I'll show you this so we pass theorem three and

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There is the M2

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So this is the one we had written over here and basically if the limit as n approaches infinity of the absolute value equals zero then we have

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The limit without the absolute value is also equal to zero. So scroll all the way back down

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Fascinating stuff. Just want to go there to see where in the video it was so way

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Far up in the video. This pretty extensive video. So let's keep going and going and now there's exercise two

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There's a theorem just double-check. So recall of theorem two right here. So limit like that

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Yes, so now what we could do is well solution one

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Let's use squeeze theorem and then our solution two. I'll use definition two

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So what we could do is so if the limit of the absolute value of a n as n approaches infinity is equal to zero

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then what we can even write is so if

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Limit as n approaches infinity a n

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Absolute value is equal to zero then

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Then what we could write is the limit as

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an approaches infinity of negative a

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Absolute value of a n this equals to we'll just use limit laws take that negative out. This becomes well

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negative now limit and

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Approaches infinity of a n she goes too. Well, this is a zero. So negative zero just equals to zero

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So yeah, that's just equals to zero and and now and

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since

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Since we have negative a n negative absolute value of a n is gonna be well less than or equal to a n

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Where this is less than or equal to the absolute value that's always make it always positive

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This one's always negative. So that's always gonna be less than or equal to the end

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So now it's gets squeezed inside. So thus

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so thus we have

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Thus limit as n approaches infinity of a n well

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This is never both of these are going to zero is going to zero is going to zero this one gets squeeze has to go to zero

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So that's limit as a limit as n approaches infinity of a n equals to zero by the

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Squeeze theorem. So yes, we see him is quite amazing

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It's very straightforward and quick. So there it is. Yeah, so now let's solve this very same theorem

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But now use solution to

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I mean look at solution to which is using a definition to which is again precise definition for sequences as

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Illustrated an exercise one as well. So we have here definition a sequence a n has this limit and we write the limit as n approaches infinity

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of a n equals l or a n approaches l as n approaches infinity if for every epsilon number epsilon grade in zero

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There's a corresponding integer and capital and such that we have we have this case if an integer if

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Any integer and greater than or I mean if all integers and greater than capital n

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Then we have the scenario the difference a n minus l the absolute values is less than this epsilon

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Then this then this is the limit. So again, so recap it also need to recap because sometimes this takes takes a while for the precise definition

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the terminology that

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Mathematics book or the calculus book uses is yeah, it appears non-intuitive, but once you dig your head into it

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Pretty yes straightforward. So if we use this definition, but in our case, so l equals to zero in our case

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Because we have the theorem is the absolute value is equal to a n equals zero

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So we're looking at that so the limit is is a zero and also we're dealing with absolute value

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So what we have is yeah, yeah, so in our case now we have this and that's right and so we have limit as

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n approaches infinity of absolute value of n of a n like this

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Equals to well zero and this is if

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n is greater than capital n then

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a

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n minus zero like a

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Epsilon we're in fact to be more complete. We'll write this as an absolute value here

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So we'll have absolute value of absolute value of a n minus l

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Like this less than epsilon, but this becomes well

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This becomes absolute value of

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Absolute value of a n minus zero so the zero just goes away

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Which equals to this one becomes well absolute value of absolute value of a and just to illustrate the point

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Which is the same thing as writing absolute value of a n

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less than epsilon

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So in other words, this is the same thing as removing the absolute value sign

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Same thing

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Yeah, so the thus this fits also for

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fits

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So this definition fits also for

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also for

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Removing absolute value sign so limit as n approaches infinity of well a n and this is equals to zero

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So yeah, if we have this is exactly the same thing

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We're gonna have if we're gonna have the exact same scenario where n is greater than n then we have this scenario of

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But instead of that we should remove that a n. So we have this exact same thing

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Yes, so thus

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We have

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Limit as n approaches infinity of absolute value of n of a n if this equals zero this also equals to the limit as n approaches infinity of

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A n like that. So yes, there's theorem and here's for completeness. Let's write the definition inside this little bubble here as well

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if n greater than n then

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We have

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a n minus l

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Less than epsilon becomes well a n minus zero

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Equals to a n like that same thing

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Less than epsilon. So yes fascinating fascinating stuff indeed. So those limits are one and the same