Limits 

Example
No. 1: MultiStage Rockets 

When a satelite is to be placed in orbit around the Earth, a multistage rocket engine is required. The reason for this is that as each stage burns out, it becomes advantageous to discard the weight of the remaining structure, thus decreasing the overall weight. When the aim is to escape Earth's gravity altogether, multistage design is not an option, but a necessity. Ideally, if we could discard unnecessary weight constantly during the flight,
we would have less and less mass to push away from Earth's gravity. What this
means is that there would be as many stages as possible. 

Our
Task 

Assuming identical materials used for each stage, as well as engine performance and characteristics, the expression used to tell how high the velocity the rocket will reach once all its fuel runs out is:  
n is the number of stages (1, 2, 3 ... ).  
r is the ratio between the entire rocket's mass and the satellite's mass, and is also a given constant (r < 1).  
is a structure coefficient, given, constant and < 1.  
is an equivalent velocity (which is a given constant).  
when  
It's hard to tell what happens to 
. Let's try to see what our TI thinks about it... 

First, type in the expression and store it in a variable called un.  
? Umm... Why did my TI return an unsimplified expression? Ans
Because you forgot a multiplcation sign beween the
and the parenthesis in the denominator. 

Now we can use the limit function to try to see what happens when 

The
general structure of the limit function is: limit(expr, var, point
[, direction]) expr is the expression we are working with, here the variable un. var is the variable in expr which we want to approach point. The direction will be discussed later on. 

As surprising as it may seem, the result indicates that we cannot achieve
an infinite velocity, even if we designed an infinite number of stages. 

This means that for every ton of satellite mass, we would require a 52 ton engine to carry it outside Earth's gravity field, 3.2 tons of which are only the structure (!). Needless to say, that in reality we never have an infinite number of stages, but only 3 (4 rarely designed). A quick calculation would reveal that for n = 3 we get:  
In other words: for every ton of satellite we would need a 69 ton engine, 4.2 tons of which are only the structure!  
Example
No. 2: Finding OneSided Limits 

This problem is to determine whether a limit exists. The given expression: 

The most direct and straightforward thing to do is ask the TI to his opinion on the matter at hand:  
Well, it seems our TI doesn't like it. Let's investigate this matter further. We know that a limit does not exist when the righthand limit is not the same as the lefthand limit. Could this be what's troubling our TI? To find a onesided limit, we simply add a parameter to the function limit. This parameter comes last, and can be positive or negative. 

For a righthand direction, it suffices to take 1
(in fact, any other positive number will do). First we try the righthand limit by adding the parameter 1 to the limit function: 

Next we check for the lefthand limit by changing the direction to 1:  
Aha! Just as we'd suspected! The onesided limits are not the same! Our TI was clever enough to check both directions and appropriately notify us that no limit actually exists.  
Example
No. 3: Finding Conditions for the Existance of A Limit for Piecewise Functions


Sometimes we may need to find the limit for a piecewise function. Cosider the following exaple: find the condition that allows for a limit of f(x) at x = 3, where f(x) is:  
How
do we go about solving this problem with our TI calculator? First, we must define
this function. We can use the when function, to define f. The when function has the following syntax: 

when(condition, true expr [, false expr [, undefined expr]])  
In other words, we are declaring that when condition
is true then the expression is true expr, otherwise if the condition
is false then false expr. If the condition
cannot be determined to be true or false, then it is undefined, in which case
we instruct to use the undefined expr (this is optional, and does
not have to be declared). This is how we define our function on the TI: 

Now all we need to do is see if there's a limit when x = 3, as requested:  
The limit does not exist, says our TI. Let's find out in detail the reason for this. First, get the righthand limit:  
Now let's see what the lefthand limit is:  
Not too surprisingly, out TI was right. Now in order for the limit to exist, we need the righthand limit to be the same as the left hand limit. Let's solve for this equaity to get a:  
And so we find out that the condition for the existance of the limit of f(x) at x = 3 is a = 2.  
______________________________________________________________________________________ Created by Andrew Cacovean, March 8, 2002 
