Comments | calculus: Limits

lyr in calculus

Limits

May 09, 2007 22:25

I've been having a lot of grief lately with limits and therefore the foundations of calculus. It's not that I don't understand what limits are. I could move on perfectly well not knowing why a limit works, just knowing the rules for evaluating them. My problem is that I don't understand the details of how and why they work. I understand the ( Read more... )

Comments 3

leprofessional May 10 2007, 02:14:42 UTC

my favorite one

lim (IQ -> 0) B.Sc. = B.A.

;)

where_was_i May 10 2007, 05:28:14 UTC

I don't know if there're actual proofs, at least at an intro level, to limits. They rest essentially on common sense and thinking about them. Think about the following, and hopefully it'll help:

Remember that you're analyzing graphs in calculus - that's all there is to it. Sometimes equations come into play, sometimes they don't, but even if you're dealing with an arbitrary equation, or a nasty equation for a graph that's too complicated to visualize easily, you're still dealing with a relationship between two variables for which you could draw a graph if you wanted to.

All a limit does is ask: where does so-and-so graph look like it's going to be at such-and-such a value?

For example limx→∞ 1/x = 0 because (thinking of the graph), y=1/x looks like it's going to towards 0 when you're very, very far out along the x-axis. That's exactly what they mean when they say "a big number over a small number." Make a table for 1/x with x values of 100, 1000, 10000, 100000, as far down as you want, and you can 'prove' to yourself the shape ( ... )

booksoverbombs May 10 2007, 07:22:58 UTC

So, here's how it goes down: by Newton's motivation of the difference quotient, the derivative of the function, say, x^2 is ((x+dx)^2-x^2)/dx, or 2x + dx. But so long as we evaluate the expression with dx != 0, the value of the derivative isn't 2x, it's 2x plus some junk. If we evaluate with dx = 0, we've committed the sin of dividing by zero, per the original expression. Berkeley raises this objection in The Analyst, perhaps one of the most pedantic essays ever written by an author with a good point in mind. It wasn't until Robinson that infinitesimals were restored to their original mathematical footing ( ... )