In this video we want to show you how to use the VARS button, which is this button right here, on the TI-84 calculator, to help us more quickly and efficiently evaluate function values. So, if you're given a certain x-value, you need to substitute that x-value into a given function to see what the output is. We can use this VARS button right here to do this fairly quickly. So, let's look at it in terms of an example. So, suppose we wanted to estimate the limit as x goes to seven of one over x minus seven. Let's just do it numerically first. So, this is a two-sided limit here. We would need to look at both one-sided limits to actually evaluate the two-sided limit. So let's say for the right-side limit we're going to evaluate these four values that are on the right side of seven but very close to seven into the given function one over x minus seven. So, what we would do first using this VARS button, I'm going to click on y equals, and I'm going to enter the function that's given. So I have to do one divided by, and I do need to use parentheses so that I make sure I'm dividing by the entire expression that's in the denominator. So I do parentheses x minus seven, and I close the parentheses if you don't use the parentheses then the minus seven would not be included in the denominator, so again parentheses are very important.
Also in this video i just want to turn your attention to these buttons down here.
These are the buttons that I'm actually pressing.
So you get a key history so if you don't follow exactly what I've done or you don't see what I'm clicking
on you'll be able to see that button down here.
Okay so I have my function entered into y1. And now I'm going to press second mode which accesses the quit feature up here, and it takes me back to the home screen and this is where I'm going to do all the calculations. So remember we want to substitute the seven point one into the function up here.
So I want to essentially substitute seven point one into y1. So to do that I can do VARS, arrow over to y-VARS and access the y-VARS menu, and from here I'm going to choose the number one function. So you can either hit enter at this point because one is highlighted, or you can hit one, and it gives you a list of all the functions that you have. These do correspond to the functions that are in the y equals feature. So I entered our function that we want to use into ysub-1. So I'm going to press one here. On the home screen then is ysub-1. I want to evaluate seven point one into that function. So I do parentheses, seven point one, and I'll close parentheses. And what this notation is going to do is take the value of seven point one for x and substitute it into the function that you have entered into y1, and give you the output. So I hit enter and we see that when we substitute seven point one into that function we get 10. Now I want to substitute in 7.01 that's over here. So in order to quickly bring up the last entry you have on the home screen, see this blue entry down here, that means that if you click second, that accesses all the blue features on the calculator, and then I click enter, it brings up my last entry. So I can now edit my last entry. So I can just arrow over and I can put in zero one close parenthesis now the calculator is going to take 7.01 and substitute it into the function I've entered into y1. We can evaluate it and find the output by clicking enter. So we now know that when I substitute in 7.01 the function's value is 100. We can follow the same process here, second enter brings up my last entry and makes it where I can edit that entry now. So I'm going to type another zero one, close the parentheses, and when we substitute 7.001 into our function to y1, we get an output of a thousand. Let's do this one more time. So I did second, enter to bring up the last entry. I'm going to edit the entry, put another zero one, close the parentheses, and you can probably guess what the output is going to be we now get 10000.
So what we see here is that the closer and closer and closer i get to positive seven on the right side,
the functions values get bigger and bigger and bigger.
So this tells me then that the limit on the right side does not exist. So the right hand limit does not exist but it does appear to be getting larger and larger.
So we would write something like this to use infinite, I'm sorry, to describe the infinite behavior
So here we've used limit notation to describe the infinite behavior that's happening here. And we can see that over here. All right now let's look at the left hand limit. So the left hand limit means I need values on the left side of seven but very close to seven and I've chosen these values here. So we want to substitute these values into the same function, so I don't need to change anything in my y1, I just need to do again second enter. It brings me up the previous entry, where I can now edit that previous entry and I'm going to type in 6.9 parentheses, I'm going to click delete up here and it will delete all those extra characters I have after that. Now I hit enter and we see that when I substitute 6.9 into the given function we get negative 10. You're probably catching on to the process now at this point, so I do second enter to bring up the last entry and make it where i can edit the entry. I type in a nine parentheses and we get negative 100. I'm going to do second enter again. I'm going to add append another nine on to that, add another digit, hit enter we get then that when I substitute 6.9 into the function that is entered into y one we get negative one thousand, and let's do this one more time.
We see that we get negative ten thousand. So now I can analyze these function values and see that as I get closer and closer and closer and closer to seven, on the left side, the function values are becoming bigger and bigger negatives if you will they're they're they're the absolute value of these functions are getting larger, they're all negative, so it looks like it's getting increasingly increasingly negative. So that means then that the left hand limit actually does not exist, but i can use infinite i can use limit notation rather to describe the infinite behavior, I have that right here. So we would say that the limit as x goes to seven is actually approaching or tends toward negative infinity. So since I have the right side limit that's tending toward positive infinity and the left side limit that's tending toward negative infinity, the actual limit then doesn't exist, and I can also not write any uh you use any limit notation to describe any infinite behavior because on one side of seven it's tending toward positive, on the other side of seven it's tending toward negative infinity. So because they're tending toward two different infinities if you will the limit simply does not exist. This was a short tutorial just to show you how you can use this VARS and y-VARS feature to calculate function values that really help you find limits numerically.