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Calculus Theorem

  1. Aug 21, 2012 #1
    This is my first course in Calculus as a Junior in college and it's already much differant then what I'm used to. I could use some help in understanding several theorems used to describe limits.

    The first states that the lim k = k as x--> aThe book describes it as being a constant function, so it's the same on either side of the limit

    The second states that the lim x = a as x-->a]. The book describes it as saying that if x approaches, so must f(x) but I don't understand that.

    If someone could explain or show me an example of these so I can understand what they're trying to say I'd really appreciate it.
  2. jcsd
  3. Aug 21, 2012 #2
    Without using the epsilon-delta definition of the limits, both can be explained by asking "what value does the function get close to when the variable x gets close to a?"

    In the first case, when f(x) = k is a constant function, the value of f(x) is always k. For example, if f(x) = 2, then f(x) = 2 for all x. Then what number is the function approaching as x approaches any number a (say 3, for example)? Well, it always is 2, so you'd expect that as x gets to 3, f(x) is still 2. In general, if f(x) = k, you'll guess that it always approaches k when x approaches a. It should be pretty obvious (you could also graph the function and see it) but you can also prove it using the limit definition quite easily.

    For f(x) = x, at any point x, the value of f at x is, well, x. This means that f(2) = 2.
    f(2.5) = 2.5
    f(2.9) = 2.9
    f(2.99999) = 2.99999
    f(3.01) = 3.01
    f(3.00001) = 3.00001

    so you can see that as x gets close to 3 from either side, f(x) gets close to 3 as well. If you wanted to see what value it approaches when x is any number a, you simply plug in values of x that are close to a, and find f(x), and you'll see where that tends to.
    Draw the graph of f(x) = x, and chose a number a on the x-axis. Then draw a rectangle around a. The values that the function takes in-between will get narrowed down as you narrow down the rectangle around a (if the limit exists, of course).

    Once again, the limit definition can prove all of this rigorously, but I'm not sure that you have seen that yet so I won't confuse you.

    The concept of a limit is very important and extremely useful in Calculus (the 2 examples here are the simplest, and they maybe seem boring because they just state the obvious. But when you'll get further in the course, you'll find much more exciting and interesting examples and applications of limits)
    Last edited: Aug 21, 2012
  4. Aug 21, 2012 #3

    Stephen Tashi

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    Perhaps you're under the impression that a "function" must be a rule or algorithm that is fairly complicated like [itex] f(x) = x^2 - 6x + 1 [/itex] or [itex] f(x) = sin(x) [/itex]. A function can be as simple as the rule [itex] f(x) = 3 [/itex] (i.e. f(x) = 3 regardless of what the value of x is.) A function can also be as simple as [itex] f(x) = x [/itex]. The rules you stated deal with these two simple cases.
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