 So in this chapter we're going to look at transcendental functions while what are they for the Purposes of this lecture. We're just going to view them as not being polynomials So let's look at some examples. We see all the trigonometric functions. They sign cosine tangent cosecant secant cotangent but then also the logarithm and the natural log and we have Exponent a Euler's number or something such as a constant to the power Variables so a to the power x or a to the power t as we've been using here Now just to show you here very quickly. These are not difficult to do to some of the basics We have the limit as t approaches some constant of the sine of t that will just be the sine of a Same goes for the cosine Same goes for exponent t there The natural log just remember that the natural log of a number is only defined for values larger than zero Simply we're just going to replace the t with a there The arc sine the arc cosine and the arc tangent Look at those remember though that the arc sine and the arc Cosine are only defined between negative one and one just think of the function the sine of t It goes up and down up and down in a periodic fashion But it never goes above one and it never goes below negative one. So it's only defined for that Let me show you why look at this the limit as t goes to zero of the arc sine of t plus one over the square root of two So yeah, instead of being of what we had before look at that. We just had the limit number. Where are we? Well, it's not shown there, but if you just look at sine and cosine there, we just have the arc sine So it is there number five We're just going to replace t with a there But yeah, we have t plus one over the square root of two if I plug in zero straight away Which is one of the techniques that we know I'm just going to be left with the arc sine of one over the square root of two and At zero at t equals zero we can see clearly the green graph they crosses the y-axis at pi over four and We remember we said the arc sine is only defined between negative one and one So look at that green graph and just stops on the left hand side and stops on the right hand side And you can see in the algebra down there negative one is less than t plus one over square root of two is less than one I'm multiplied by square root of two throughout and then I just subtract one from either side and I'm left with t Alone in the middle and now you can see why t is only defined as far as that green graph is concerned So that was quite easy Look at this one the limit as t goes to zero of t to the t times e to the power negative three t plus one over two t squared plus t and If we have a look at the graph it does seem something happens at zero the graph certainly crosses the y-axis there And it looks like between two and a half and three maybe two point seven somewhere there If I were to just to plug in t equals zero into my expression there, I'm going to get zero over zero now later in chapter eight I'll show you another way to deal with that kind of problem But yeah, I just note that in the denominator at least I can take t out as a common factor Which will cancel out t in the numerator and denominator for me And now I'm left with the limit as t goes to zero of e to the power negative three t plus one over two t plus one and if I were to replace t with zero now I'm left with just well as number which is two point seven something something just as we see in the graph there no problem Look at this though What happens to this functions sine and cosine of t as t approaches positive or negative infinity well We know we said sign and cosines is this periodic function that just goes up and down up and down It does not settle on a specific y-axis value So there is no limit the limit does not exist as far as sine and cosine are concerned as T goes to either positive or negative infinity. It doesn't never settles down to a specific value What happens to the tangent? Well, what happens to tangent as t goes to you see their n times pi plus pi over two So they certainly this periodicity of n pi Let's make n zero now So just t goes to pi over two and as you can see clearly the red graph goes up to Positive infinity on the y-axis. So certainly infinity is not in number So you can't say that the graph settles on a certain value on the y-axis. It will climb forever Therefore that limit does not exist if I make n one so t goes to pi plus pi over two that's three Pi over two you see that will happen again. The red graph will go up to infinity And you'll also see that approaching that vertical asymptote, which is what we'll call it From left and right hand side. It's either going to be positive infinity from the left Negative infinity from the right for all these vertical asymptotes The arc tangent of T is a beautiful graph though. There you see it Does settle on a value it does not climb forever and ever this is one you can remember certainly That is going to settle get closer and closer and closer for whatever number you have a Larger and larger number value of t as as you try to approach infinity approach infinity It's only going to settle on pi over two. So that's a beautiful one to remember usually with these Transcendental functions, it's difficult to solve some of them and we're going to make use of the infamous squeeze theorem or sandwich Sandwich theorem ugly words for something in mathematics, but that's what we have and it's a very easy rule If you look at the three graphs at the bottom right at the bottom You'll see negative t squared That's the red one at the bottom the red one at the top is t squared and somewhere in between we have half t squared And if we look at a certain interval on the t-axis here say from Negative two to positive two that orange line all the y values on that orange line fall between the two red ones There's not one that jumps out Now I can have I'm just wondering you you can have a piecewise defined function Which will say that that orange graph at zero might be ten That doesn't matter Because it will tend to stay between those two So if I take a limit that means where the where the orange graph tends to go They just suddenly squeezed between the other two So if I can work out the limit as t approaches zero of the top red one And I can work out the limit as t approaches zero of the bottom red one That'll be exactly the same as the limit as t approaches zero of the orange graph. That's all that's going on there So let's have a look at this g of t is our red one now wiggling up and down up and down up and down It's t squared Times sine of one over t. What happens is t approaches zero if I were just to plug in zero there I'm gonna have zero times the sine of one over zero. That's just not on That's just not on but I see that it is constrained from the top and the bottom We have the blue graph there t squared the red graph negative t squared We can certainly work out that the limit as t approaches zero of those two is zero Therefore the limit of t squared times sine of one over t will also be zero Now the thing that you're going to encounter a lot in the transcendental functions are the piecewise to find transcendental functions Here we have the limit as t approaches zero of t squared plus two times the cosine of t plus two when t is less than zero so look on your t-axis To the left of y equals zero Or t equals zero I should say there's our rate graph So as we approach t equals zero from the left We're dealing with the top of top part of the piecewise function the red graph there and at the bottom We approaching it from the right and we can clearly see in this piecewise to find functions the left hand side and the right hand side Are not equal if you just were to plug in zero into the values there you're gonna see From the left hand side, it's four from the right hand side. It's negative three So this limit does not exist. There's a left-hand limit. There's a right-hand limit, but they are not the same One interesting limit to memorize you'll see it all over the show probably in your tests in your exams The limit is t approaches zero of sine t over t and that's just one you can clearly see on the red graph That's one one way the only way that we have to solve that now because we have to plug it in between zero over zero I said in chapter eight We're gonna look at another way to solve this problem But for now we can one of the techniques that we can do is just plug in values closer Make t closer and closer and closer to zero and you can clearly see the f of t Tries then which a sine of t over t Tries to go to one you can clearly see that in the two tables at the bottom So the sine of t over t is one is a beautiful other way to solve that But you can just memorize the fact that the sine of t over t equals one the same goes for t over the sine of t The limit there is t approaches zero is also going to be one and you can see in the red graph there clearly it's one Last problem here the limit is t approaches zero of sine squared of t over t squared We make use of our limit laws We see that sine squared t over t squared can be written as sine over t times sine over t Limit laws as far as the product of two functions are concerned. So I'm doing the two limits separately I know that the limit is t approaches zero of the sine of t over t is one So this is going to be one times one and the solution is one So there you go the basics of transcendental functions and limits