Trigonometric limits & squeeze theorem
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Limits of trigonometric functions
- [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.