To the point where one can't continue... (not with 5 mph!)
a.k.a The LIMIT!
This page will review several important concepts and go over some basics of the limit. For a comprehensive study of limits, please consult the documents for download. Also, in the documents, there will be complete lists of formulas and theorems. (the documents are due July)
Page Walkthrough
1. What is a limit?
2. The deltaepsilon proof
3. a way to think about limits
4. The messy types of 0/0 and Inf/inf
5. the squeeze theorem
2. The deltaepsilon proof
3. a way to think about limits
4. The messy types of 0/0 and Inf/inf
5. the squeeze theorem
What is a limit?The limit of a function f(x), at some value a, is simply, what the value of the function approaches when x gets close to a.
Of course, most of the times, the limit is equal to the value of f(a). (What else would you expect?) Most of the times. There are exceptions. Those exceptions are what make the concept of the limit tricky and timeconsuming. Let's take a moment to think what kind of function would it take to make the limit different from the value of the function. Of course, polynomial functions who are defined everywhere, like y=x, are easy prey to our mathematical minds. The limits of those functions at any x is equal to the value of the function at that point. Right. Here, I present a part of the FUNCTION BLACKLIST, whose contents produce messy limits, but sadly, takes up most of the space in calculus tests. 1. Functions who are not defined at a. There is no value at a to talk of. (e.g. Rational functions with the denominator as zero at the point x=a.) 2. Rational functions whose numerator or denominator appears to march off to infinity. Infinity allows no limit. 3. Functions who vibrate violently around the value a. 4. Functions who approach different values when x gets close to a from the left side and the right side. Be extra careful when encountered with such functions. The limit might not appear to be what it is. Here, I offer a way that will allow thinking about limits a lot easier:
Think of the graph of the function y=f(x) as a roller coaster track. You, as an unlucky amusement park visitor, is thrown into an eternally powered roller coaster that rides on the graph. The limit of the function at x=a, is what your position is when the shadow of your cart goes over x=a. Of course, if the track is solid at that point, your cart is on the track. However, if there is a small hole on the track that is, the function is not defined at x=a, but defined at both sides of it, your cart will fly right over the gap hold your breath. So if the function has the form y=2x, yet x is not defined at 2, then the limit of the function at x=2 is still four. (By the way, for those who play MW3, Mind the Gap is my favorite campaign mission.) Now, if the limit goes to infinity then you visit the kingdom on the clouds. We say the limit diverges, or it is infinity. Just like the U.S. debt ceiling, if you know what I'm talking about. If there are wild variations the cart will get thrown off track the limit does not exist! Using the same reasoning, we can see if the function approaches different values from both sides, your cart, alongside with you in it, will undergo a peculiar experience, not available at any licensed amusement park. Chances are you are not going to wake up to tell the tale. The limit does not exist. Think of that as the graph of a function
Squeeze TheoremInteresting fact: The textbooks in China (for the international audience) and many other countries translate this into "the sandwich theorem", which is an even more vivid way of describing the essence of this principle.
This principle simply says, if f(x)<=g(x)<=h(x) for all x, then for any value a, the limit of f(x), g(x), h(x) as x approach a follows the rule: lim f(x)<=lim g(x)<=lim h(x). This, if we take a moment of thought, is obvious. Consider the three curves in the plane. The curve of f(x) always stays above the curve of g(x), and the curve of g(x) always stay above the curve of h(x). And the limit? Think of it as a prison. h(x) and f(x) are the guards, guarding g(x) in between. of course, g(x) could not go outside of the bounds defined by the two guards. (assuming ideal "unbribable" guards, that sadly, only exist in the kingdom of mathematics) Squeezed to nothingness...

For the sake of our conscience, and to suppress the demands of those instructorsA clean conscience is the evidence of a bad memory.
The Delta Epsilon ProofWe have mentioned before that limits are what functions approach when the independent variable gets close to a certain value. Here is the problem as any obnoxious high school math teacher would put it please proooove it.
We will succumb to the demands for once. The proof for a limit is called the "deltaepsilon" proof. The key point is to find a certain positive number delta, so that when x is within delta of a number a, the value of the function f(x), must not diverge from the limit at x=a (suppose it's L) by another positive number epsilon.
We can write as: when xa<delta, yL<epsilon. The key to the proof is to relate delta with epsilon. You are not seriously thinking of giving a table of epsilons and then making another table of deltas, and assuming they can continue forever, are you? If so, you have fallen into a common trap calculus, I tell you now, is half about continuing forever. The other half? Presenting infinity in a finite amount. That is the hard part. That is the interesting part. For example, how do we prove that the limit of the function y=2x when x is 2 is 4? We need to find a delta, so that when x is within delta of two, y is within epsilon of 4. We need to relate delta to epsilon, so no matter what an epsilon you give me, I can find a corresponding delta. The solution here is that delta is equal to half of epsilon. (Take a moment to think about why) So if you require y to be within 2 of 4, I say x is within 1 of 2. Note that most of the times, delta is smaller, and much smaller than epsilon. However, there is one such delta. If you can't find the certain delta, there are two possibilities: 1. The limit does not exist. 2. You need to revise your algebra. ContinuityWhat is continuity? Simply speaking, it's whether or not the graph of a function is smooth.
There is not really much to say about continuity, save for the fact that it lays the foundation for limits, therefore for derivatives and integrals, and calculus alike it's definition is as follows: A function f(x) is continuous at x=a if and only if the limit of f(x) at x=a is equal to f(a). Simple enough. 0/0 limits and limits at infinityPre L' Hopital's Rule
When evaluating limits of rational functions where both the numerator and the denominator equal 0 at the point x=a, without the help of L' Hopital's Rule, much algebraic manipulation is needed. We have to get either the numerator or the denominator to equal to something other than 0. One way to do this is to divide both by x or a power of x, multiply both by x or by a power of x. Or divide both by the factor in the denominator or numerator that makes it zero. For example, for the limit of (x+3)(x3)/(x3)(x+5), try dividing both by x3. Remember we are dangling our feet in the water x is not technically 3. It is just arbitrarily close to it! So we can do the indicated simplification! Now, what do we do when something in the numerator or denominator decides to go to infinity as x=a? No Problem. Use division or multiplication to cancel that term out, or bring that term to the denominator. Remember! (any number)/infinity=0! So, we can calculate the limit of the function (2x+2)/(2x^2+1) at x=infinity as follows: 1. Divide numerator and denominator by x. 2. Now the function's numerator does not contain any x's. This means it is a number other than infinity. The denominator, on the other hand, contains a big term 2x. 3. We safely conclude the limit is 0. GENERAL RULE: Most of the times, if both the numerator and the denominator of a rational function is a polynomial, then when x approaches infinity 1. if the numerator contains a higher power of x, the limit is infinity. 2. if the denominator contains a higher power of x, the limit is zero. Post L' Hopital's Rule Things become much more simple. Just note two things: 1. It is senseless to compute the limit of the derivative using L' Hopital's Rule, as the rule uses derivatives itself. 2. Don't keep differentiating! STOP when either the numerator or the denominator ceases to be zero! 