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# Limits of trigonometric functions

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.D (LO)
,
LIM‑1.D.1 (EK)

## Video transcript

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 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 0 and we can 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 it so if you think in degrees this was a 45 degree angle and in general if I'm dealing with a sine or 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 there 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 want to 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 0 over negative 1 which is completely fine if it was negative 1 over 0 would be in trouble but this is just going to be equal to 0 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 sine of X over cosine of X now sine 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 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 is 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 one over tangent of X it's cosine of X over sine of X it's a little bit this is limit as X approaches PI of this and is PI in the domain of cotangent of X well though 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 take 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
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