Limits of trigonometric functions | Limits and continuity | AP Calculus AB | Khan Academy

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'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 sine of x, because it's continuous and it's 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.

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 the sine and cosine of.

If you think in degrees, this is a 45° 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.

I can make a similar statement about cosine of x. The 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 of tangent of X, which is sin of X over cosine of x.

Both of these are defined for pi, and so we could just substitute pi into the function. We just want to ensure that we don't get a zero in the denominator because that would make it undefined. So we get sin of pi over cosine of pi, which is equal to 0 over -1. This is completely fine; if it was negative 1 over 0, we would be in trouble. But this is just going to be equal to zero, so that works out.

If I were to ask you what is the limit as X approaches Pi / 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 / 2 of sin of X over cosine of X. Now, sine of Pi / 2 is 1, but cosine of Pi / 2 is zero. So, if you were to just substitute it in, this would give you 1 over 0.

One way to think about it is that Pi / 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. A way to deduce that is that Pi / 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/ 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 is 1 / tangent of X; it's cosine of x over sin of X.

This is the 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 -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. Once again, this is not in the domain of that, and so there's a good chance that we have no limit. When the point we're taking the limit to is in the domain of the trigonometric function, we're going to have a defined limit. 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.