Should You Get a Cat or Dog? Are You a Cat person or a Dog person? What pet should you get?
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Introduction to Limits | ZJ learning | Limits#1 - Duration: 5:36.
One of the most important and widely used topic in math is calculus. Today we will begin
our journey to the world of Calculus. We will start off with one of the pre-requisites for
this topic - Limits.
First of all lets take a general limit notation, The way to pronounce this is limit of f(x),
as x tends to a, is equal to L What this actually means is that the value
of the function f(x) will get closer and closer to L, when x becomes closer and closer to
'a'. However remember that here x is not equal to a.By finding the limit of a function we
only want to know what happens to the value of the function when x approaches to 'a' and
not when x = a.
To understand this better consider this example. Here we have the function y = x squared. When
x = 2, y = 4. Lets consider a small increment 'delta' from 'x'.
So when x = 2 - 'delta', y1 = (2 - 'delta') whole thing squared So that will give us 4
- 4*delta + 'delta' squared. Let this be equation 1
When x = 2 + 'delta', y2 = (2 + 'delta') whole thing squared. That is equal to 4 + 4*delta
+ 'delta' squared. Let this be equation 2 In equation 1 and 2, when 'delta' is deleted
that is as delta tends to zero, we get y1 = 4 and y2 = 4
We can show this operation using our limit notation.
So here we have the left hand limit, limit of (x squared) as x tends to 2 - 'delta'.
So this is the left hand limit. So that would be equal (2 - 'delta') whole thing squared
giving us 4 - 4*delta + 'delta' squared. So for the 2nd equation we have the right hand
limit. So this is limit of (x squared) as x tends
to 2 + 'delta' giving us (2 + 'delta') whole thing squared. That give us 4 + 4*delta +
'delta' squared. So this will be equation 2.
When delta tends to zero, Equation 1 and 2 become equal
So the limit of (x squared) as x tends to 2 = 4.
So here since the left hand limit and the right hand limit are equal when 'delta' is
deleted it means that the limit exists. So basically what has happened here is that
the 'x' coordinate 2 - 'delta' is squeezed from left to right and 2 + 'delta' is squeezed
from right to left. Similarly y1 and y2 are sandwiched to '4'. So this method of calculating
limits is known as the method of squeezing.
So lets consider this function. So here we have a function y = f(x)
So if we take the right hand limit of this, the limit of f(x) as x tends to zero, here
the positive sign means that we are approaching it from the right hand side. So as we approach
zero from the right hand side we get the limit as 1.If we find the left hand limit of f(x)
as 'x' approaches zero, here the negative sign means that we are approaching from the
left hand side. So as we approach zero from the left, we get the value -1.
So here the left hand limit and the right hand limit are not equal therefore the limit
does not exist. So therefore we can conclude that the limit
of f(x) when 'x' tends to 'a' = L only when the limit of f(x) when x tends to 'a positive'
= L and when limit of f(x) when x tends to 'a negative' = L. That is the left hand limit
and the right hand limit must give us the same value.
So that was our basic introduction to limits. In our next video we will learn about the
properties of limits and how to answer questions related to limits.
If you have any questions, please ask us in the comments section below. Until we meet
again take care
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Mourinho tipped for sensational Chelsea return if Sarri is axed - Duration: 1:46.
Jose Mourinho could return to Chelsea for a third spell if Maurizio Sarri is axed
That is the view of talkSPORT presenter , who believes the pressure on Sarri is increasing following the midweek thrashing at Bournemouth
Chelsea were beaten 4-0 on the south coast in what was their heaviest defeat for 23 years
Sarri kept his players behind in the dressing room for over an hour before admitting he may be unable to motivate his squad
His comments led to suggestions that Sarri wouldn't be at Stamford Bridge for the start of next season
And Durham believes Chelsea should turn to one of their former managers - or another club legend
"Who next? Because Jose Mourinho is free, you could see it happening couldn't you
Of course you can see it happening," he said. "He's available, and the other one is Frank Lampard
When you look at what Man United have done, you brought up Ole Gunnar Solskjaer, if you bring back a club legend and it fits like a glove
"He's already had a little taste of it at Derby where things are going very well
"Can you see Frank Lampard managing Chelsea, taking over from Sarri? I don't think he'll be able to turn it down
"
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Favorite Products (January) 2019 - Duration: 5:19.
Hi everyone! My name is Karen. Welcome back to my channel and I'm so excited
cause I'm going to share with you my favorite products for the month of
January this year. But before that if you are new here don't forget to Subscribe,
click the bell to turn on notification so that you can be notified of my upcoming
videos. And also please don't forget to like this video and my other videos on
my channel if you have watched it and now let's move on to my favorite
products for the month of January. The first one is this. This is basically a
coin bank. I chose this product because of the design, it is so nice I love the
color combination of this one. It has pink, blue, brown, green, and the print is
really nice it has an animal print on it. And the design is perfect for those who
want to have attractive coin bank and I also chose to save this year
because I wanted to follow my New Year, New Year resolution which is to save
money. It doesn't mean that I don't save money often but this time
is different because I wanted really to follow my budget
and limit my spending habit. I wanted to make sure that I have a financially,
financially independent year as of this time. And so I chose this product as my
favorite product for this month because I wanted to remind myself that it is
really important to save money before spending. Okay my second favorite product
is this one. This is actually a product of Vice Ganda. I've checked her website
and I found this one the Lip and Cheek tint and the color is red it
is Kyondi shade and I really like it because it's perfect to mix with and
blend with my Maybelline Fierce Fuchsia. This - so I am wearing the Lip tint
right now I haven't tried the cheek tint yet but when I've tried it I'll share
it later. And I really like the lipstick because the combination is
perfect it has shade of pink and red which is
really nice and the price is 145 pesos and this is a gift from my
husband so I love my choice. I've been trying to buy
a lipstick from them but I only had it last December when my husband arrived
from Manila so this is one of the products that I'll use this coming
February because of the shade it has red and has pink. The
third favorite product for the month of January is this one. Here I know this is
just so simple it's just a mug, but I'm going to use this to remind myself to
eat and prepare healthy food. I'm going to use this for my blended fruits and
vegetables, fresh ones, for breakfast. So I learned that I really
have to eat a lot of fruits and vegetables in order to have healthy skin
and healthy body. I don't want to experience sickness I want to make sure
that I am healthy. And I chose this mug it's because of the design and it's not
a typical one it is so long and it has this a print a flower one. It has this
lining of black and it's a combination of white, blue, and pink so you know that I am
a pink lover so this one is perfect. Okay so last but not the least, fourth
favorite product for the month of January is this one,
so this is basically a makeup setting spray and this is from Pro Studio. This
is so important for me because last year I have never used a setting
spray before when I apply makeup but this time I used this twice and my friends said
that my face is better compared to the time that I'm not using a setting spray
aside from I'm Mary Kay products so combined with this one this is so
perfect. And I I can't remember the price I'll just place it here, and I bought
this at Beauty MNL last December and this is the last item that I'm going
to use when I apply my makeup. There you have it, my favorite products for the
month of January 2019. If you liked this video, don't forget to click thumbs up,
don't forget to subscribe to my channel and click the notification bell.
And see you next time. Bye!
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