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So what are limits? Limit is a term in mathematics whose meaning
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is approach. When we put limit notation to the left of a function,
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what we mean is we are asking "what does this function approach
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as the input value approaches a number?" So, for example, when we write down the limit x to
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0 of sin x divided by x, that means we are asking what does sin x divided by x approach
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as the x value approaches a number?" or more specifically, what value
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does sin x divided by x approaches as the x value approaches 0? The illustration goes like
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this, imagine that there is someone named Siska, Siska wants to buy shoes at Shop A.
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On the way, someone asks Siska,
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"Siska, where are you going?" then Siska replied, "I am going to Shop A".
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Even though it's possible that when Siska arrives at Shop A, it turns out that the shop is closed, but that doesn't
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matter because what the person asked earlier is the destination that Siska wants to go to, which is Shop A.
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So, whatever happens at Shop A actually doesn't affect the answer to the
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person's question, the most important thing is that Siska's destination is clearly Shop A. So, this is
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the concept of limits. Limit asks about the value that is approached by a function, where
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here Siska is the function and the person asking earlier is us as the observer.
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Therefore, the best way to answer the limit question is to use a test.
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So, let's try plugging in the values of x that are closer to 0, and then we'll see what value
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does sin x to divided by x approach. So, for example, if we enter the value of x = 0.1, why do we choose
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0.1? Because 0.1 is close to 0. So, sin 0.1 divided by 0.1 is = 0.998334.... and so on.
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Note, angle units in trigonometry limits are always
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radian units, very rarely degrees are used. And by the way here you can just
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use a calculator, because sin 0.1 is quite difficult to calculate manually. Okay,
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then we try to enter the value of x again which is even closer to 0, that is x = 0.01
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then sin 0.01 divided by 0.01 equals to 0.99983.... and so on. Let's try even closer input,
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x = 0.001 then the result is 0.99999.... and so on. Watch the pattern! It turns out that the closer
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the value of x is to 0, the value sin x divided by x approaches is 1. So, we write down the
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limit x to 0 of sin x divided by x equals 1. Intuitively we can translate this
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as: if the value of x approaches 0 then the value approached by sin x divided by x is 1.
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Actually, there is something interesting here, the exact value of sin x divided by x for x = 0 is actually
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undefined. What is the exact value? The extract value is the value we get
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when we actually enter the value of x which is equal to the number in question. So, the
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exact value of the function f(x) = sin x divided by x for x = 0 is f(0) = sin 0 divided by 0. We know that sin 0
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equals 0, so sin 0 divided by 0 gives 0/0 which is undefined.
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Intuitively what this means is: if the value of x is actually 0,
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then the value of sin x divided by x is undefined. And this is interesting, right? The function is the same, that is
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sin x divided by x, but the limit value and the exact value are different. And in fact, if
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we look at the graph of the function sin x divided by x, the graph has a hole at the point x = 0,
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which indicates that it is not defined there. But even so, from this graph
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it is clear that if the value of x approaches 0, the value that is approached by sin x divided by x is 1.
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So, this is the difference between the exact value and the limit value. Sometimes
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there are some functions whose exact value and limit value are not the same. And this is very reasonable,
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because the exact value and the limit value are indeed two different concepts. But of course
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there are lots of other functions where the limit value and the exact value are the same, like this one,
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namely g(x) = x to the power of 3, the exact value and the limit value are always the same regardless of the value of x.
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So in conclusion, what is a limit? Limit means approaches. If you understand the meaning of the term
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approach, then that means you already understand what a limit is. Then the second
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important thing is that you have to be able to distinguish what is the difference between the limit value and the exact value? The limit value
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is the value that a function arpproaches as the input value goes to a number,
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while the extract value is the value of a function when the input is equal to a number.
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If you can understand up to here, then that means you already understand the essence of the concept of limit. Now
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you can continue studying the next chapter of Calculus, which is about Derivatives and Integrals.
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In the next episode, we will discuss "When Do Limits Exist and When Do Limits Not Exist?",
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because in my opinion this is quite important to know. However, you can go straight
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to episode 3, which is about Derivatives. You will still understand without any problems. So, make sure
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you LIKE this video and SUBSCRIBE so you don't miss it. See you in the next video!
