Limits

Today we will be looking at limits using Mathematica. Mathematica is a great tool where we can graph and find the limits of functions. First start by opening up Mathematica on your computers. Go to “Search Windows” on the bottom left corner of your desktop. Start Typing in “Mathematica” It should pop up and you can click on it to open the program.
	Great now to start your going to click this button  Then you are going to select “Plain Text”. Then you can start typing so we are going to call this “Limits” So go ahead and type that. Then you are going to highlight it and then right click, then you will select “style” and then “Title”. This will change your font and color of your words to look more like a title of the document. Now click on “Palettes” and Click “Classroom Assistant” this tool box will help us throughout today’s lesson.
	We know in mathematics, a limit is the value that a function or sequence “approaches” as the input or index approaches some value. So lets input some examples. To in put an expression you will need to type Limit[(your function), x->0] so lets try this function. Your arrow can be found on the classroom assistant.
	Limit[Sin[x]/x,x->0] then press Shift Enter. Your screen should look like this .  If you would like your function to look like this ((Sin[x])/(x^2)) you can use the classroom assistant. Next you can first define your function then calculate the limit. First lets define the function x Log[x]. So type into a new line and hit Shift Enter. Then to use this function to find the limit you will type then press Shift Enter again and you should get 0. 
	Some functions have no definite limit. For these the output will be an interval, which is where the uncertain value lies between the interval of ( x-min, x-max). For example type in this function and hit Shift Enter and you should get an interval response. .
	Lets plot a function. Lets plot the function (1/x) to do this you can click Basic commands on your classroom assistant then 2D the Plot and you can input the information. Or you can simply type it in as shown on a new line. you should see this after hitting Shift Enter .
                       									
We can see that depending on the direction we choose to look to to the limit is different. If we would like to approach the function from above we can type (limit as it approaches 0 from above) we see the limit is infinity. And if we want to do from the other direction we can type(limit as it approaches 0 from below)  we see that the output is negative infinity. 
	Let’s define the function as f[x] so first delete the previous f[x] we defined above. You can do this quickly by clicking the brackets on the right side of your screen.Select the brackets that contain the function f[x] and press delete. Then just write  then press shift enter.
 Now as we input random numbers we will get an output you can plug 2 in like this :You can plug in numbers if you would like to play around with it.
 	 Next lets plug in 4 and see what happens we get: . This is where we would need to find the limit. We type in  then press Shift Enter and you should get ((-5)/8). To get more of a visual on this function lets Plot it. 
 I put 100 for the x max just to get a good visual of the graph. You can play around with it and input 10, 5 or other numbers.
	 
	When typing a graph and you need to change the y axis you need to type PlotRange-> then you y min and max. You can also find this in your classroom assistant under Basic Commands->2D then Option: Automatic Positioning 
	Lets try another function h[x]= ((2(x^3)-1)/(5(x^3)+x-1))->∞ First lets define h[x] by typing in Now find the limit of h[x] and press shift enter(the infinity button is on your classroom assistant).  Once you press shift enter you should get 2/5. Once again lets get a better view on this function we do this by plotting it: Plot[h[x],{x,-1,100}]
	  when we look at this function we can see that the limit is 2/5. 
	Lets look at another function.

k[x_]:=(((x^2)+4x-12)/((x^2)-2x))

k[10]

k[100]

k[-10]

k[-200]

We can see that by plugging in numbers that as x goes to infinity, y gets closer and closer to 1. Lets see if we are correct.

Limit[k[x],x->∞]

Limit[k[x],x->-∞]

We are correct!

Plot[{(((x^2)+4x-12)/((x^2)-2x)),1},{x,-10,10},PlotStyle->{Automatic,Dashed},PlotRange->{-10,10}]

This adds a Dashed line so that we can see the function approaching the limit of 1.

Plot[Sin[x]/x,{x,-1,1},Epilog->{PointSize[Large],Point[{0,1}]}]

We see here that we were able to put a point at the max.

Here are some problems to try on your own:
	 Limit[(((((6+x))^2)-36)/x),x->0] =
	
	 
	Limit[((6+4x)/((x^2)+1)),x->-3]=
	
	
	Limit[8-3x+12(x^2),x->2]=
	
	Try Plotting these functions too!

The Mathematica inputs.

Limit[Sin[x]/x,x->0]

Limit[Sin[1/x],x->0]

g[x_]:=x Log[x]

Limit[g[x],x->0]

f[x_]:=((x-(Sqrt[3x+4]))/(4-x))

h[x_]:=((2(x^3)-1)/(5(x^3)+x-1))

Plot[1/x,{x,-1,1}]

Limit[1/x,x->0,Direction->1]

Limit[1/x,x->0,Direction->-1]

f[2]

f[450]

f[4]

Limit[f[x],x->4]

Plot[f[x],{x,-1,100}, PlotRange->{-1,1}]

Limit[h[x],x->∞]

Plot[h[x],{x,-1,50}]

Practice answers:

Limit[8-3x+12(x^2),x->2]

Limit[((6+4x)/((x^2)+1)),x->-3]

Limit[(((((6+x))^2)-36)/x),x->0]

Limit[(((x^2)+2x-3)/((Abs x-1))),x->1,Direction->1]

Limit[(((x^2)+2x-3)/((Abs x-1))),x->1,Direction->-1]