Pratica di Shadowing: The Limit (do not use L'Hospital rule) - Impara a parlare inglese con i video

Okay, let's talk about the limit as theta goes to 0 of sine theta over theta.

Yes, we put 0 right here and right here, we end up with 0 over 0.

So maybe let's just go ahead and do the Lafittos rule.

Let's put on d, d theta, d, d theta.

So we are showing work, and our teacher will be happy, right?

So this is the limit as theta goes to 0.

And then when you differentiate sine theta, we get cosine theta.

And then when we differentiate theta, we just get 1.

And then now we can just put the 0 right here.

And in the end we are saying this is just cosine of zero.

Of course we can work this out.

This is just one.

That's it.

Took us about like 37 seconds.

That's it. And yeah.

Okay.

Has this ever happened to you guys?

You did this on the test.

But in the end you end up with a big X like that.

And you end up with zero points, zero marks for this question.

If this has happened to you, leave a comment down below.

And you might be wondering, what's so wrong with this?

Yes, we do the L'AbiDoc rule, and this is a 0 over 0 situation.

Is it because I didn't show work?

I forgot to put down this is a 0 over 0 situation at Lh.

Maybe I just forgot to show work.

Okay, now here's the problem.

The issue is that when you differentiate sine theta, Yes, the answer is cosine theta, but how do you know that?

Well, if you guys look back to the proof of the derivative of sine theta,

in fact, you need to first know what the limit as theta goes to 0 of sine theta over theta is first, before we can talk about the derivative of sine theta.

So, in other words, we cannot use derivative, we cannot use Lapidus' rule to figure this limit out, because otherwise, this is what we call the circular reasoning.

That's really bad, right?

Therefore, we must use another way to figure this out, and let's do it right here.

The way that we actually need to do it is by using geometry.

And first of all, let's just talk about we want to get the limit as theta goes to 0 plus.

And of course, let's put down side theta over theta.

Well, you'll see this is actually what we need, because later on we can also use this to figure out the limit as theta goes to 0 minus.

And you'll see you'll end up with the same answer, and then we can conclude the theta goes to 0 that limit.

Well, what we need to do first is, we are going to begin by looking at the unit circle.

And because we know theta is going to 0+, that means the angle is small and it's positive.

So we just need to consider the first quadrant.

So let me just draw the picture right here for you guys.

Let's say this right here is my unit circle.

And because unit circle, so we know this right here would be 1.

since the angle is small, so let me just put this down right here.

Right?

I just need a first quadrate.

That's all.

Now, I know this is one, of course.

Okay, next what we need is, let's first look at this arc.

And on the circle, we know that the arc has a formula, right?

The R-length formula.

We just need two things.

First, we need the R, the radius, and then next we need the angle theta in radians.

And of course, we are all doting out this angle theta is in radians already.

So in this case, I just need to do 1, which is the radius, times the radians theta angle right here.

So in other words, the arc length is just theta.

This represents the angle, this right here represents the length.

So that's the first thing that we need.

Next, what we'll do is, I will look at the inside and right here,

starting at this point, I'm going to draw a line straight down so that we can form a right angle.

I will also figure out a formula for this length.

And you can see that we have a right triangle now.

If you look at this right triangle, we do know the hypotenuse, and this will be the opposite side.

So let me just indicate this for you guys right here.

we have the opposite, that's the opposite, and then the hyponius is just 1.

And let me just indicate, over hyponius, and hyponius is 1.

Opposite over hyponius is what?

Yes, it's just sine, and of course we have that angle.

So therefore, this length is just sine theta.

Alright, we're making good progress.

Next, I will be looking at this point right here, which is 1 comma 0, and I will draw a line straight up.

In fact, this is also the tangent line.

You'll see tangent line.

Alright, so you see this right here, but in fact I'm not going to draw all the way, because what I will do is, I'm going to actually extend this right here, so I'll just put on some dot dot dot,

and then I will stop right here, alright, when they meet.

Well, in this case, I can create another right triangle But the problem is,

I don't know how big this is So I cannot use sign right here anymore, unfortunately, right?

But it's okay, because we know from here to here is 1

So if you just look at this bigger right triangle here This right here is once again the opposite

but this time this right here is the adjacent that we know

so just put on a d j

which in this case is also one what's opposite for adjacent yes it's tangent

so right here as told you guys earlier in fact that's the tangent line

so we get this right here equal to tangent theta like that cool Cool, huh?

Now, we have three things to compare.

The green length and then the red curve and also the blue length, right?

The blue line.

Well of course this right here is the smallest, this right here is the second, this right here is the biggest, right?

Because you see pretty much this right here is between of this and that.

From the picture you can see that's the biggest.

So let me just put this down right here for you guys.

First of all we know sine theta which is this length right here is the smallest.

Next, we have the r length, which is the theta, and then tangent theta is the biggest.

And, of course, we can put down some inequality.

It goes like this.

Well, theta changes.

In fact, theta is approaching zero plus.

So, in fact, let me also include the equal sign right here.

100

This is still true, isn't it?

101

Now, don't forget what we are trying to achieve.

102

This is what we have.

103

We want to achieve sine theta over theta, but we only have this.

104

But it seems like theta should be in the denominator.

105

So we are going to take the reciprocal of all this.

106

And notice, everything right here, they are all positive because we have the theta is just a positive small number, right?

107

So when I take the reciprocal, I get 1 over sine theta.

108

And because everything is positive, you can just flip the inequality.

109

And you get this right here, and then this right here is 1 over theta.

110

And then you can just flip that as well.

111

If you have 1 over tangent, that's just cotangent.

112

In another word, you get cosine theta over sine theta.

113

Well, well, I want to have the sine theta on the top, so why don't we just multiply everybody by sine theta right here?

114

And when we do that, this times this is, of course, just 1, and then, of course, sine theta is positive.

115

When you multiply by a positive number, the inequality stays.

116

So I'll just put down sine theta on the top over theta, and then this times that, of course, the sine theta cancel we have this right here,

117

which is just cosine theta.

118

So this is what we have.

119

What's next?

120

Well, I'm trying to figure out the limit.

121

From this inequality, I'll just take the limit as theta approaching 0 plus.

122

So I will do it here.

123

And then I will also do it here, the limit as theta approaching 0 plus.

124

And then, of course, we also do it here.

125

Right?

126

Cool.

127

This right here, 1 is always 1.

128

So this is 1.

129

You maintain the inequality.

130

And then, this is exactly what we're trying to find out.

131

I don't know yet, so let me just write this down.

132

The limit as theta approaching 0 plus sin theta over theta.

133

And then, this right here is what?

134

You put 0 plus right here, cos 0 plus.

135

Yes, we did it over there, but this was not legit.

136

You get one.

137

Now, you see, this limit is in between one and one.

138

Therefore, we can use the squeeze theorem to argue that.

139

So I'll just put this down by the squeeze theorem.

140

Yes, T stands for stiff as well.

141

right?

142

Anyway, we can conclude that the limit as theta approaching zero plus of sine theta over theta,

143

this right here, has to be equal to 1 as well, isn't it?

144

And as you can see right here, we are done, and we got this right here.

145

Cool.

146

Yes.

147

Now, we are not done yet.

148

We also have to look at the zero minus version.

149

So let me just put this down right here for you guys real quick.

150

If you take a look of the limit as theta approaching 0 minus of sine theta over theta.

151

Well, this right here is actually not that bad at all.

152

Remember, when we have 0 plus, this is as saying, you know, theta is like about 0.0001.

153

When you have 0 minus, it's just a negative of that version, pretty much, right?

154

So, what you can do is, and depending on how much we're going to show,

155

You can just look at this as sine of 0 minus, and then on the bottom you have 0 minus like this.

156

Well, sine is an odd function.

157

This right here is a negative number.

158

So this right here, as I say, you can just take the negative on the outside because sine is an odd function.

159

So you have negative of sine, and the input now, it will be the positive version, namely 0 plus.

160

Likewise, 0 minus is as a negative of the 0 plus.

161

Negative, negative, cancel.

162

This right here gives you 1 because of that.

163

It depends on how much work you have to show, but this right here, hopefully, is a good enough argument for many of you guys.

164

And now, as we know, when we have the right limit and also the left limit equal to the same number,

165

we can conclude that When we have the limit as theta goes to just zero of sine theta over theta,

166

this right here has to be one.

167

Finally, we did it.

168

This is what I call the limit.

169

This right here is the limit.

170

Why?

171

Because it happens a lot.

172

Because this right here also happens a lot.

173

It hurts a lot of students.

174

And once again, leave a comment down below and let us know

175

if this happened to you if you did that before or not.

176

And in fact, I do feel guilty because sometimes I do this myself as well but I do it in my head.

177

So as long as you don't write it down, we are good.

178

Anyway, hopefully you guys all like this and let me know

179

if you guys have any questions and if you guys are new to my channel, be sure to subscribe.

180

You guys can find a lot more interesting my videos on my channel and thank you guys so much for that.

181

And as always, that's it.

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Vocabolario e note di pronuncia per questa lezione

Questo video contiene 181 frasi e 2020 parole da ripetere con lo shadowing. Il parlato dura 12:07. Chi parla ha un ritmo naturale di circa 167 parole al minuto, vicino a una conversazione quotidiana. Il 86% delle parole rientra nelle 3.000 più comuni dell’inglese; conviene studiare le altre prima di iniziare.

Vocaboli chiave di questo video

15 parole del video che vale la pena imparare, con pronuncia e significato:

Parola	Pronuncia	Significato
angle sostantivo	/ˈæŋ.ɡəl/	angolo
minus	/ˈmaɪ.nəs/	meno
inequality sostantivo	/ˌɪn.ɪˈkwɑ.lɪ.ti/	disuguaglianza
triangle sostantivo	/ˈtɹaɪˌæŋɡəl/	triangolo
formula sostantivo	/ˈfɔː.mjʊ.lə/	formula
conclude verbo	/kənˈkluːd/	finire, concludere
derivative sostantivo	/dɪˈɹɪvətɪv/	derivato, derivata
indicate verbo	/ˈɪndɪˌkeɪt/	indicare
depend verbo	/dɪˈpɛnd/	dipendere
flip verbo	/flɪp/	capovolgere, spadellare
cancel verbo	/ˈkæn.sl̩/	depennare, cancellare
adjacent aggettivo	/əˈd͡ʒeɪ.sənt/	adiacente
squeeze verbo	/skwiːz/	spremere, stringere
radius sostantivo	/ˈɹeɪ.di.əs/	radio
theorem sostantivo	/ˈθiərəm/	teorema

I phrasal verb che sentirai

Parola	Pronuncia	Significato
figure out verbo		scoprire, rendersi conto
find out verbo		scoprire
get over verbo		superare, vincere
go ahead verbo	/ˌɡoʊ əˈhɛd/	procedere, andare avanti

Pronuncia a cui fare attenzione

Chi parla usa 24 contrazioni e forme ridotte, come don't, I'll, I'm. Pronunciale nella forma breve, così come le senti.

I suoni “th”: theorem /ˈθiərəm/, theta /ˈθiːtə/
Parole lunghe — attenzione all’accento: inequality /ˌɪn.ɪˈkwɑ.lɪ.ti/, derivative /dɪˈɹɪvətɪv/, differentiate /dɪf.əˈɹɛn.ʃi.eɪt/, geometry /d͡ʒiˈɑ.mə.tɹi/, reciprocal /ɹɪˈsɪpɹək(ə)l/

Come esercitarsi con questo video

Ascolta tutto il video una volta senza parlare e annota le parole che non conosci.
Inizia a velocità 0,75×, fai shadowing frase per frase e torna alla velocità normale quando diventa facile.
Registrati e confronta con l’originale, facendo attenzione a parole come angle, minus, inequality.

La grammatica di questo video

Le strutture che chi parla usa di più, con le parole esatte del video:

Struttura	Nel video
Frasi relative who / which + frase — un’informazione in più su una persona o una cosa	here, which is · theta which is · length, which is
Forma passiva be + participio passato — conta ciò che accade, non chi lo fa	are done · are not done

Cos'è la tecnica dello Shadowing?

Shadowing è una tecnica di apprendimento delle lingue supportata da studi scientifici, originariamente sviluppata per la formazione dei traduttori professionisti e resa popolare dal poliglotta Dr. Alexander Arguelles. Il metodo è semplice ma potente: ascolti un audio in inglese di madrelingua e lo ripeti immediatamente ad alta voce — come un'ombra che segue il parlante con un ritardo di solo 1–2 secondi. A differenza dell'ascolto passivo o degli esercizi di grammatica, lo shadowing costringe il tuo cervello e i muscoli della bocca a elaborare e riprodurre simultaneamente i modelli di discorso reale. La ricerca dimostra che migliora significativamente la precisione della pronuncia, l'intonazione, il ritmo, il discorso connesso, la comprensione dell'ascolto e la fluidità del parlato — rendendolo uno dei metodi più efficaci per la preparazione alla prova di speaking dell'IELTS e per la comunicazione reale in inglese.

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