Shadowing Practice: The Limit (do not use L'Hospital rule) - Learn English Speaking with Video
בחר מצב תרגול
כל מצב פותח חדר תרגול במסך מלא.
181 משפטים
אם המשפטים קצרים או ארוכים מדי, לחץ על ערוך כדי להתאים אותם.
1
Okay, let's talk about the limit as theta goes to 0 of sine theta over theta.
2
Yes, we put 0 right here and right here, we end up with 0 over 0.
3
So maybe let's just go ahead and do the Lafittos rule.
4
Let's put on d, d theta, d, d theta.
5
So we are showing work, and our teacher will be happy, right?
6
So this is the limit as theta goes to 0.
7
And then when you differentiate sine theta, we get cosine theta.
8
And then when we differentiate theta, we just get 1.
9
And then now we can just put the 0 right here.
10
And in the end we are saying this is just cosine of zero.
11
Of course we can work this out.
12
This is just one.
13
That's it.
14
Took us about like 37 seconds.
15
That's it. And yeah.
16
Okay.
17
Has this ever happened to you guys?
18
You did this on the test.
19
But in the end you end up with a big X like that.
20
And you end up with zero points, zero marks for this question.
21
If this has happened to you, leave a comment down below.
22
And you might be wondering, what's so wrong with this?
23
Yes, we do the L'AbiDoc rule, and this is a 0 over 0 situation.
24
Is it because I didn't show work?
25
I forgot to put down this is a 0 over 0 situation at Lh.
26
Maybe I just forgot to show work.
27
Okay, now here's the problem.
28
The issue is that when you differentiate sine theta, Yes, the answer is cosine theta, but how do you know that?
29
Well, if you guys look back to the proof of the derivative of sine theta,
30
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.
31
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.
32
That's really bad, right?
33
Therefore, we must use another way to figure this out, and let's do it right here.
34
The way that we actually need to do it is by using geometry.
35
And first of all, let's just talk about we want to get the limit as theta goes to 0 plus.
36
And of course, let's put down side theta over theta.
37
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.
38
And you'll see you'll end up with the same answer, and then we can conclude the theta goes to 0 that limit.
39
Well, what we need to do first is, we are going to begin by looking at the unit circle.
40
And because we know theta is going to 0+, that means the angle is small and it's positive.
41
So we just need to consider the first quadrant.
42
So let me just draw the picture right here for you guys.
43
Let's say this right here is my unit circle.
44
And because unit circle, so we know this right here would be 1.
45
since the angle is small, so let me just put this down right here.
46
Right?
47
I just need a first quadrate.
48
That's all.
49
Now, I know this is one, of course.
50
Okay, next what we need is, let's first look at this arc.
51
And on the circle, we know that the arc has a formula, right?
52
The R-length formula.
53
We just need two things.
54
First, we need the R, the radius, and then next we need the angle theta in radians.
55
And of course, we are all doting out this angle theta is in radians already.
56
So in this case, I just need to do 1, which is the radius, times the radians theta angle right here.
57
So in other words, the arc length is just theta.
58
This represents the angle, this right here represents the length.
59
So that's the first thing that we need.
60
Next, what we'll do is, I will look at the inside and right here,
61
starting at this point, I'm going to draw a line straight down so that we can form a right angle.
62
I will also figure out a formula for this length.
63
And you can see that we have a right triangle now.
64
If you look at this right triangle, we do know the hypotenuse, and this will be the opposite side.
65
So let me just indicate this for you guys right here.
66
we have the opposite, that's the opposite, and then the hyponius is just 1.
67
And let me just indicate, over hyponius, and hyponius is 1.
68
Opposite over hyponius is what?
69
Yes, it's just sine, and of course we have that angle.
70
So therefore, this length is just sine theta.
71
Alright, we're making good progress.
72
Next, I will be looking at this point right here, which is 1 comma 0, and I will draw a line straight up.
73
In fact, this is also the tangent line.
74
You'll see tangent line.
75
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,
76
and then I will stop right here, alright, when they meet.
77
Well, in this case, I can create another right triangle But the problem is,
78
I don't know how big this is So I cannot use sign right here anymore, unfortunately, right?
79
But it's okay, because we know from here to here is 1
80
So if you just look at this bigger right triangle here This right here is once again the opposite
81
but this time this right here is the adjacent that we know
82
so just put on a d j
83
which in this case is also one what's opposite for adjacent yes it's tangent
84
so right here as told you guys earlier in fact that's the tangent line
85
so we get this right here equal to tangent theta like that cool Cool, huh?
86
Now, we have three things to compare.
87
The green length and then the red curve and also the blue length, right?
88
The blue line.
89
Well of course this right here is the smallest, this right here is the second, this right here is the biggest, right?
90
Because you see pretty much this right here is between of this and that.
91
From the picture you can see that's the biggest.
92
So let me just put this down right here for you guys.
93
First of all we know sine theta which is this length right here is the smallest.
94
Next, we have the r length, which is the theta, and then tangent theta is the biggest.
95
And, of course, we can put down some inequality.
96
It goes like this.
97
Well, theta changes.
98
In fact, theta is approaching zero plus.
99
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.
הורד אפליקציה
כל מה שאתה צריך כדי לדבר בשטף
הגיית AIציון לכל משפט
תרגול IPAשלוט בכל צליל
אוצר מיליםבנה את מאגר המילים שלך
משחק אוצר מיליםלמד תוך כדי משחק
FOR YOU
המלץ על סרטון
C181 משפטים
Mathematician Collapses All Functions to One Weird Formula
Daily ConversationUniversity StudentsWorking Adults
C12
64 משפטים
GCSE Physics - How to Draw Ray Diagrams - Concave & Convex Lenses (2027/28 exams)
Daily ConversationHigh School StudentsUniversity Students
C12
82 משפטים
CAP Theorem Simplified
Daily ConversationUniversity StudentsWorking Adults
C11
575 משפטים
Lec-6 Simplex Method | Maximization Problem | Mathematical Example Solution | Operations Research
Daily ConversationUniversity Students
C12
138 משפטים
Derivatives: Crash Course Physics #2
Daily ConversationUniversity Students
B2215 משפטים
PTE SWT: One-Sentence Formula for 79+
Daily ConversationUniversity StudentsWorking Adults
C1552 משפטים
The Professor Who Taught People How To Think (1962)
Daily ConversationUniversity StudentsWorking Adults
B21
72 משפטים
Fight brain-rot with Worked Examples
Daily ConversationUniversity StudentsWorking Adults
C11
54 משפטים
Traceroute: You're Probably Reading the Results Wrong
Daily ConversationUniversity StudentsWorking Adults
B2158 משפטים
7 Cấp Độ Toán Học: Bạn Đang Ở Đâu?
Daily ConversationUniversity Students
B21
29 משפטים
OHM's Law in 3 minutes (Beginner friendly)
Daily ConversationUniversity Students
C16
104 משפטים
How real are the real numbers, really?
Daily ConversationUniversity Students
אוצר מילים והערות הגייה לשיעור הזה
המילים שחוזרות הכי הרבה: theta, sine, limit, plus, length. בסרטון הזה יש 181 משפטים ו-2020 מילים לתרגול Shadowing. הדיבור נמשך 12:07. הדובר מדבר בקצב טבעי של כ-167 מילים בדקה, קרוב לשיחה יומיומית. 86% מהמילים נמצאות בין 3,000 המילים הנפוצות ביותר באנגלית; כדאי ללמוד את השאר לפני שמתחילים.
אוצר מילים מרכזי בסרטון הזה
15 המילים המתקדמות ביותר בסרטון, עם ההגייה והמשמעות שלהן:
| מילה | הגייה | משמעות |
|---|---|---|
| theta שם עצם | /ˈθiːtə/ | טֵיתָא |
| sine שם עצם | /saɪn/ | סינוס |
| tangent שם עצם | /ˈtæn.d͡ʒənt/ | טנגנס |
| minus | /ˈmaɪ.nəs/ | פחות, מינוס |
| cosine שם עצם | /ˈkoʊ.saɪn/ | קוסינוס |
| inequality שם עצם | /ˌɪn.ɪˈkwɑ.lɪ.ti/ | אי־שוויון, אִי־שִׁוְיוֹן |
| triangle שם עצם | /ˈtɹaɪˌæŋɡəl/ | משולש \ מְשֻׁלָּשׁ |
| reciprocal שם תואר | /ɹɪˈsɪpɹək(ə)l/ | הדדי |
| theorem שם עצם | /ˈθiərəm/ | מִשְׁפָּט |
| cotangent שם עצם | /ˈkoʊˌtænd͡ʒənt/ | קוטנגנס |
| denominator שם עצם | /dɪˈnɒmɪneɪtə(ɹ)/ | מְכַנֶה |
| hypotenuse שם עצם | /haɪˈpɑtənus/ | יתר |
| geometry שם עצם | /d͡ʒiˈɑ.mə.tɹi/ | גֵּאוֹמֶטְרְיָה |
| subscribe פועל | /səbˈskɹaɪb/ | נִמְנָה |
| likewise תואר הפועל | /ˈlaɪkˌwaɪz/ | כְּמוֹ כֵן |
Phrasal verbs שתשמעו
| מילה | משמעות |
|---|---|
| figure out פועל | הֵבִין, מצא תשובה |
משפטים שכדאי לחזור עליהם
משפטים קצרים ושלמים מהסרטון שאפשר להשתמש בהם בשיחה יומיומית:
- Let's put on d, d theta, d, d theta.
- Has this ever happened to you guys?
- Is it because I didn't show work?
- Let's say this right here is my unit circle.
- Alright, we're making good progress.
הגייה ששווה לשים לב אליה
הדובר משתמש ב-24 קיצורים וצורות מקוצרות, כמו don't, I'll, I'm. אמרו אותם בצורה הקצרה, כפי שאתם שומעים אותם.
- הצלילים של “th”: theta /ˈθiːtə/, theorem /ˈθiərəm/
- מילים ארוכות — שימו לב להטעמה: inequality /ˌɪn.ɪˈkwɑ.lɪ.ti/, differentiate /dɪf.əˈɹɛn.ʃi.eɪt/, derivative /dɪˈɹɪvətɪv/, reciprocal /ɹɪˈsɪpɹək(ə)l/, denominator /dɪˈnɒmɪneɪtə(ɹ)/
צלילים שדוברי עברית מתקשים בהם:
- /w/ — שפתיים מעוגלות, לא /v/: inequality /ˌɪn.ɪˈkwɑ.lɪ.ti/, squeeze /skwiːz/, quadrant /ˈkwɑ.dɹənt/, quadrate /ˈkwɑd.ɹət/, likewise /ˈlaɪkˌwaɪz/
- /æ/ — פתוח יותר מה-“אֶ” העברית, לא כמו “a” ב-“cup”: tangent /ˈtæn.d͡ʒənt/, triangle /ˈtɹaɪˌæŋɡəl/, cotangent /ˈkoʊˌtænd͡ʒənt/
- /r/ — הלשון לא נוגעת בשום מקום, לא ה-“ר” הגרונית: triangle /ˈtɹaɪˌæŋɡəl/, radian /ˈɹeɪdiən/, differentiate /dɪf.əˈɹɛn.ʃi.eɪt/, derivative /dɪˈɹɪvətɪv/, reciprocal /ɹɪˈsɪpɹək(ə)l/
איך לתרגל עם הסרטון הזה
- הקשיבו לכל הסרטון פעם אחת בלי לדבר, ורשמו את המילים שאתם לא מכירים.
- התחילו במהירות 0.75×, תרגלו Shadowing משפט אחר משפט, וחזרו למהירות רגילה כשזה מרגיש קל.
- הקליטו את עצמכם והשוו למקור, תוך תשומת לב למילים כמו theta, sine, tangent.
מהי טכניקת השאדואינג?
שאדואינג היא טכניקת למידת שפה מבוססת מדע שפותחה במקור להכשרת מתורגמנים מקצועיים ופופולרית על ידי הפוליגלוט ד"ר אלכסנדר ארגואלס. השיטה פשוטה אך עוצמתית: אתה מקשיב לאודיו באנגלית ילידית וחוזר עליו בקול רם מיד — כמו צל העוקב אחרי הדובר עם עיכוב של 1–2 שניות בלבד. בניגוד להאזנה פסיבית או תרגילי דקדוק, שאדואינג מכריח את המוח והשרירים בפה לעבד ולשחזר דפוסי דיבור אמיתיים בו זמנית. מחקרים מראים שזה משפר משמעותית את דיוק ההגייה, האינטונציה, הקצב, הדיבור המחובר, הבנת השמיעה ושטף הדיבור — מה שהופך אותו לאחת השיטות היעילות ביותר להכנה לIELTS Speaking ולתקשורת באנגלית בעולם האמיתי.