lim  f(x) = n
      x->c

That is to say, "as x gets arbitrarily close to c, the value of f goes to n (or gets arbitrarily close)." Take note that we are not concerned with what the function's value is at c but instead the value of the function as x gets extremely close to it. If you wanted the value at c it would be as easy as plugging the number into the function (we will see that limits are a little harder). However, since we want the function to be smooth we need to check the values of the function very close to c to make sure that we do not get some way off numbers. To show this let's look at an example. Say we were given the following function: f(x) = x . Nothing difficult about that function. It is just a line that goes through the origin at a forty-five degree angle. Let's say we wanted to evaluate this limit of the function as x goes to zero. The limit would also be equal to zero right? Now consider this pricewise-defined function:

This function is also a line for most of the domain, but when x equals zero f suddenly jumps up to five. What would the limit of this function be as x goes to zero? It is still zero! That is because we do not care what f equals at zero but what it equals as x gets really close to it.

Hopefully, with the above example and the example I gave at the beginning (the parabola and line pricewise-defined function), you will see a certain flaw in how I have explained things so far. When looking back at the examples you should take note that we are not considering from what side x is coming. The limit we have been dealing with so far is called a one-sided limit, meaning it does not take into consideration which side you are coming from. This is something very important. The side from which the limit is being evaluated from is denoted by a superscript after c like this:

      lim  f(x) = n    // from the left side -- Left-hand Limit
     x->c-

      lim  f(x) = n    // from the right side -- Right-hand Limit
     x->c+

Now let's look at our two above examples as limits coming from two different sides. In the first pricewise-defined function you can see that the limit is different from each side as x gets closer to zero. Specifically, as x comes from the right to zero the limit is two. However, if you were to come from the left the limit would be zero. Remember that we are not concerned with what the function's value is at zero but instead really close to it so, it does not matter that the parabola part of the function is not defined at zero. In the second example you should quickly see that the limit from each side is the same, once again due to the fact that we do not care what the function's value is at zero.

Computing Limits and General Rules.

So, now that you have been introduced to the notion of limits. Now we will learn some general rules for computing them and then try a couple of examples.

First the easiest kind of a limit: the limit of a constant function. In general:

      lim  n = n
      x->c

That is merely stating that the limit of a constant function (for example: y=3) is always equal to the constant.

The second easiest limit to evaluate is that of polynomial functions. You will soon learn to love polynomial functions. They are continuous at all points in the domain (smooth), they are easy to differentiate, and easy to integrate. You can evaluate the limit of a polynomial by simply plugging in c for x and solving:

      lim  f(x) = f(c)
      x->c

The next few rules for limits have you break up your main limit into smaller ones and evaluate them separately, then putting them all together. For example, if you had this: y = x^2 + x . All we need to really do here is evaluate the limit of (x^2) and (x) and then add the two limits. This is given by:

      lim  [ f(x) + g(x) ] = lim  f(x) + lim  g(x)
      x->c                   x->c        x->c


        // and if you can do it with addition, of course you
           can do it will subtraction


      lim  [ f(x) - g(x) ] = lim  f(x) - lim  g(x)
      x->c                   x->c        x->c

We can also use the same idea and rules for computing limits of two functions multiplied together or divided by each other:

      lim  [ f(x) * g(x) ] = lim  f(x) * lim  g(x)
      x->c                   x->c        x->c


                    lim  f(x)
            f(x)    x->c
      lim   ---- =  ---------         // remember that you cannot divide by zero though
      x->c  g(x)    lim  g(x)
                    x->c

And even further, you can use the same idea for computing the limit of function that has been raised to a power [example: y = (x+1)^3] by evaluated the limit of the function first and then raising it to the power. In general:

      lim  [ f(x)^n ] =  [ lim  f(x)]^n
      x->c                 x->c

These are the basic rules you should know. Usually you can be given an seemingly complex limit to evaluate and all you have to do is break it down into a bunch of simple limits. Let's look at some examples. First, I'm going to give an example from each rule and then I'll give you a big limit that uses all the rules. Here we go:

      Rule One

         lim  5 = 5
         x->2

      Rule Two and Three

         lim  3x^3 - x^2 + 2x - 1 = 3(3)^3 - 3^2 + 2(3) - 1 = 77
         x->3

      Rule Three

         lim  (3x+1)(2x-1) = lim  (3x+1) * lim  (2x-1) = 7 * 3 = 21
         x->2                x->2          x->2

      Rule Four

                       lim  (x-1)
                x-1    x->3            2
         lim   ----- = -----------  = --- = 1/5
         x->3  x^2+1   lim  (x^2+1)    10
                       x->3

      All the rules
                                             /   lim  (3x^2 - x + 1)     \ 3
               /  3x^2 - x + 1   \ 3         |   x->4                    |
         lim   |---------------- |   =  lim  | ------------------------- |
         x->4  \(x^2 + 1)(2x - 3)/      x->4 | lim  (x^2+1) * lim  (2x-3)|
                                             \ x->4           x->4       /

                /  45  \3   / 45 \3    91125
         = lim  | ---- |  = | -- |  =  ------ = (approx.) .148
           x->4 \ 17*5 /    \ 85 /     614125

Well, that last example was a really bad one because the answer is a crazy fraction but you should get the idea. Your first order of business when dealing with limits should be to break it down to smaller limits. However, sometimes these rules are not good enough. For example:

      lim   x^2 + 4
      x->2  -------
             x - 2

If we were to use the methods we have learned so far we would get a division by zero. So, we must find another way. In this case you should factor out (x^2 + 4) into the multiplication of two binomials. With that done it is sometimes possible to cancel a few terms out a get away from dividing from zero. Let's use the example from above:

      lim   x^2 + 4     (x-2)(x+2)
      x->2  -------  =  ----------
             x - 2         x - 2

        // the (x-2)'s in the numerator and denominator cancel out and you are left with:

      lim  x + 2 = 4
      x->2

So, the above rule, in a general form, is to factor the numerator into two (or more) binomials and reduce the fraction by canceling out terms. But, even that does not work all the time. For example:

              _
      lim   \/x - 2
      x->4  -------
             x - 4

The limit of the denominator is once again zero and factoring the numerator would give you two crazy binomials which would not even be able to cancel out. This time we multiply the numerator and denominator by the conjugate of the numerator:

              _         _
      lim   \/x - 2   \/x + 2           x - 4
      x->4  ------- * -------  =  ---------------
             x - 4    \/x + 2     (x - 4)(\/x + 2)


         // The (x-4)'s cancel out


      lim      1        1
      x->4  -------  = ---  =  .25
            \/x + 2     4

Continuity

Enough with computing limits. I believe you have enough tools for finding the limits of pretty much any function so we will go on to continuity and then we will be done. Continuity is a property functions can have which determines if a function is smooth or not. Simply put, a function is continuous if you could draw it on a piece of graph paper without having to lift your pencil. So, if we were to refer all the way back to the beginning, that pricewise-defined function of a parabola and a line would not be continuous because you would have to lift your pencil at the origin and move up to draw the line.

An entire function is continuous if the function is continuous at every point in its domain. A function is continuous at one point if the one-sided limit at that point is equal to the value of the function at that point. In general:

      If:

         lim  f(x) = f(c)
         x->c

      Then the function is continuous at point [c, f(c)]

Now the only thing keeping us back from checking if a point on a graph is continuous is evaluating the one-sided limit. First you must evaluate the left and right-side limits. If those two limits equal each other then that is what the one-sided limit is. If the two limits are not equal to each other then the one-sided limit does not exist. So, if we were to rewrite the above to give a better definition:

      If:

         lim   f(x) = lim   f(x) = f(c)
         x->c-        x->c+

      Then the function is continuous at point [c, f(c)]

Most functions are continuous. The only ones you usually need to worry about are pricewise-defined functions. Before we try a few of those out first some functions that are always continuous. Polynomials are continuous by definition. Rational functions are continuous in all of its domain except when the denominator is zero. Trigonometric functions are also continuous except at points where the domain is undefined [for example: tan (pi/2)]. Also, in addition to those functions, the sum, difference, product, and quotient of any two (or more) continuous functions are continuous at all parts of the domain except when the denominator is zero.

Those functions we just reviewed cover pretty much every type of function there is. The functions that will give you trouble are pricewise-defined functions. Let's say we were give the following function:

      lim   f(x) = 0^2 + 3 = 3
      x->0-

      lim   f(x) = -0^2 + 3 = 3
      x->0+

So, we have found that the limit of the function, coming from both sides of (x = 0), are equal to each other. This does not necessarily mean that the function is continuous at (x = 0) though. If you remember one the pricewise-defined functions I gave at the beginning where the function was defined as a line throughout the entire domain except at one point where the graph shifted suddenly up on the y-axis. You should see that the limit from both sides at that point are equal however the graph is clearly not continuous at that point. Thus, after you check if both the two-sided limits are equal you must then check if they are equal to the value of that function at that point:

     lim  f(x) = 3
     x->0


     lim  f(x) = f(0)  ?
     x->0


        // since f(x) is defined at (x = 0) with this equation (-x^2 + 3) we use it
           to evaluate f(0)


     f(0) = -0^2 + 3 = 3

Therefore the function at that point inherits all the attributes of being continuous. It has the same two-sided limit at the point and the limits equal the value of the function at the point. Also, since the pricewise-defined function is made up of two continuous functions the entire function is continuous. Before getting too comfortable with the notion continuity take into consideration that the above problem is rather trivial. The problems can get more difficult. But, as long as you can compute limits you should have no problem.

The end ... and for those who feel lost ...

We are done with limits and continuity! This turned out to be much shorter than I thought. But, for some reason I feel I may have lost a few people along the way. Although I believe that if you read over the tutorial again you probably get it let me just tell you that limits are not an absolute necessity for understanding derivatives. Even though a derivative is a limit I plan on introducing derivatives the limit way and the non-limit way (but you will see that they are pretty much the same). So do not feel bad if some of this went over your head because derivatives are the important things. In fact, I have read some very good calculus books that never even mentioned the word "limit".

That is it for now. Thanks for reading.

- Brandon Williams