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

• is think about limits involving trigonometric functions.

• 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.

• 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

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