Limits

[lyr]

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

Comments

[where_was_i]

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 of 1/x. y=1/x gets ridiculously close to 0 as x gets bigger and bigger, but of course it never reaches zero. I tend to think of the smooth shape of the graph instead of the 'steps' from trying 1/100, 1/1000, 1/10000 etc, but it's nice to see it once. One of the cool things about math is that a function 'means' only one thing, but you can analyze it many different ways =).

Now for something like limx→0 1/x does not exist, you can play the same game. On the right hand side of zero (x=0.1, 0.01, 0.001, 0.0001, etc.) the value of 1/x gets huge - the graph climbs up the y-axis. Of course on the left hand side (x=-0.1, -0.01, -0.001, -0.0001, etc.), the y=1/x values are negative, and it climbs down the y-axis. Now it makes no sense to ask what value it 'looks' like y=1/x is going to be at 0, because it has two different values (+∞ and -∞). Hence, it does not exist.

This is the basic idea, and I just used y=1/x as an example, but it's the same thing for any function. For example: limx→-1 √(x) does not exist (in the real numbers) because there's nothing to the left of zero on the graph y=√(x) (the function is undefined for x<0, to say it another way).

For infinitesimals, I invoke the idea of analyzing a graph, again. Hopefully you learned the idea of slope way back in the day, where m=Δy/Δx. Again, think of a graph. You're looking at a right triangle with one point as (x1,y1) and the other as (x2,y2). For a differential, or infinitesimal distance, say 'dx,' all you do is pretend that the point x1 and x2 are getting really, really, really close to each other (infinitely close, actually). As for notation, as you 'push' the points x1 and x2 close to each other, you go from the 'Δ' to the 'd'.

I don't know of any place that you can find this explained online. I came up with this on my own as I learned calculus and as I pieced together the cryptic things I read in books, and the slightly more elegant, but still burdensomely technical things teachers have told me. I'd suggest playing around with everything on your own, and thinking about it. Try not to get lost in the notation and try to grasp what is going on instead of how it is going on (look for the underlying theme, not the details). Just by asking questions, I'd say you're off to a good start. Hope this was helpful =).