[0:00]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 sin of x. Pause the video and see if you can figure this out. Well, with both sin of x and cosine of x, they're 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 sin of X, because it's continuous and is defined at sin of pi, we would say that this is the same thing as sin of pi. And sin of pi, you might already know is equal to zero. And 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 4 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 going to be cosine of pi over 4. And that is going to be equal to square root of 2 over 2. This is one of those useful angles to know what the sine and cosine of it's if you think in 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'll 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 sin 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 want to ensure that we don't get a zero in the denominator because that would make it undefined.
[3:07]So we get sin of pi over cosine of pi, which is equal to 0 over negative one, which is completely fine. If it was negative 1 over 0, we'd be in trouble. But this is just going to be equal to zero. So that works out. But if I were to ask you what is the limit as X approaches pi over 2 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 2 of sin of X over cosine of X. Now sin of pi over 2 is 1, but cosine of pi over 2 is 0. So if you were to just substitute it in, this would give you 1 over 0. And it one way to think about it is pi over 2 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 sine, cosine, tangent or cosecant, secant or cotangent, if we're taking a limit to a point that is 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 a way to deduce that is that pi over 2 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 2. 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, it's 1 over tangent of X, it's cosine of X over sine of X. It's a limit, 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 to just substitute pi in, you're going to get negative 1 over 0. 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. So 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.
