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  • - [Instructor] What we're going to do in this video

  • is think about limits involving trigonometric functions.

  • So let's just start with a fairly straightforward one.

  • Let's find the limit as x approaches pi of sine of x.

  • Pause the video and see if you can figure this out.

  • Well, with both sine of x and cosine of x,

  • they are defined for all real numbers,

  • so their domain is all real numbers.

  • You can put any real number in here for x

  • and it will give you an output.

  • It is defined.

  • And they are also continuous over their entire domain,

  • in fact, all of the trigonometric functions are continuous

  • over their entire domain.

  • And so for sine of x, because it's continuous,

  • and is defined at sine of pi,

  • we would say that this is the same thing as

  • sine of pi,

  • and sine of pi, you might already know,

  • is equal to zero.

  • Now we could do a similar exercise with cosine of x,

  • so if I were to say what's the limit as x approaches,

  • I'll just take an arbitrary angle, x approaches pi over four

  • of cosine of x?

  • Well once again,

  • cosine of x is defined for all real numbers,

  • x can be any real number.

  • It's also continuous.

  • So for cosine of x, this limit is just gonna be cosine

  • of pi over four,

  • and that is going to be equal

  • to square root of two over two.

  • This is one of those useful angles

  • to know the sine and cosine of.

  • If you're thinking degrees, this is a 45 degree angle.

  • And in general, if I'm dealing with a sine or a cosine,

  • the limit as x approaches a of sine of x

  • is equal to sine of a.

  • Once again, this is going to be true for any a,

  • any real number a.

  • And I can make a similar statement about cosine of x.

  • Limit as x approaches a of cosine of x

  • is equal to cosine of a.

  • Now, I've been saying it over and over,

  • that's because both of their domains are all real numbers,

  • they are defined for all real numbers that you put in,

  • and they're continuous on their entire domain.

  • But now, let's do slightly more involved

  • trigonometric functions, or ones that aren't defined

  • for all real numbers, that their domains are constrained

  • just a little bit more.

  • So let's say if we were to take the limit

  • as x approaches pi of tangent of x.

  • What is this going to be equal to?

  • Well, this is the same thing as the limit

  • as x approaches pi.

  • Tangent of x is sine of x over cosine of x.

  • And so both of these are defined for pi

  • and so we could just substitute pi in.

  • And we just wanna ensure that we don't get a zero

  • in the denominator, because that would make it undefined.

  • So we get sine of pi

  • over cosine of pi which is equal to zero over negative one,

  • which is completely fine.

  • If it was negative one over zero, we'd be in trouble.

  • But this is just gonna be equal to zero.

  • So that works out.

  • But if I were to ask you, what is the limit

  • as x approaches pi over two of tangent of x?

  • Pause the video and try to work that out.

  • Well, think about it.

  • This is the limit as x approaches pi over two

  • of sine of x over cosine of x.

  • Now sine of pi over two is one,

  • but cosine of pi over two is zero.

  • So if you were to just substitute in,

  • this would give you one over zero.

  • And one way to think about it is pi over two

  • is not in the domain of tangent of x.

  • And so this limit actually turns out, it doesn't exist.

  • In general, if we're dealing with the sine,

  • cosine, tangent, or cosecant, secant, or cotangent,

  • if we're taking a limit to a point

  • that's in their domain, then the value of the limit

  • is going to be the same thing as the value

  • of the function at that point.

  • If you're taking a limit to a point

  • that's not in their domain,

  • there's a good chance that we're not going to have a limit.

  • So here, there is no limit.

  • And the way to do that is that pi over two

  • is not in tangent of x's domain.

  • If you were to graph tan of x, you would see

  • a vertical asymptote at pi over two.

  • Let's do one more of these.

  • So let's say the limit as x approaches pi of cotangent of x,

  • pause the video and see if you can figure out

  • what that's going to be.

  • Well, one way to think about it,

  • cotangent of x is one over tangent of x,

  • it's cosine of x over sine of x.

  • This is a limit as x approaches pi

  • of this.

  • And is pi in the domain of cotangent of x?

  • Well, no, if you were just to substitute pi in,

  • you're gonna get negative one over zero.

  • And so that is not in the domain of cotangent of x.

  • If you were to plot it, you would see a vertical asymptote

  • right over there.

  • And so we have no limit.

  • We have no limit.

  • So once again, this is not in the domain of that,

  • and so good chance that we have no limit.

  • When the thing we're taking the limit to

  • is in the domain of the trigonometric function,

  • we're going to have a defined limit.

  • And sine and cosine in particular are defined

  • for all real numbers and they're continuous

  • over all real numbers.

  • So you take the limit to anything for them,

  • it's going to be defined and it's going to be

  • the value of the function at that point.

- [Instructor] What we're going to do in this video

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B2 中高級 美國腔

1.53 三角函數的極限(Limits of trigonometric functions | Limits and continuity | AP Calculus AB | Khan Academy)

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    yukang920108 發佈於 2022 年 07 月 02 日
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