|Mar5-12, 08:17 PM||#1|
The "importance" of Limits
I am a student in the physics and engineering fields. I have been doing calculus for two years. I understand that the limit is, in a sense, the "building block" of calculus. Differentiation and the derivative is defined by calculating the difference quotient Δy/Δx of a function and taking the limit as Δx approaches 0. Definite integration involves finding the area of the region under a function using n number of rectangles and letting n approach infinity. Again, I understand that limits are important because they are framework for calculus.
Finding the derivative of a function using the limit process is great for demonstrating the nature of differentiation, but this can be a tedious process. There are proven methods for computing derivatives; There is the power rule, product and quotient rules, the chain rule, all based on the properties of limits.
Finding the area underneath a function using the limit process, again shows how the area can primitively be solved. Needless to say, this is also a tedious (and paper consuming) process. Use the FTC or integration by substitution/parts.
Is it vitally important to memorize the properties of limits themselves? Learning them initially was great; they were intuitive and simple to understand, but when I ventured into the deeper parts of calculus, I found myself having to, every now and then, review seeming useless theorems and rules.
After a while, limits just seem to be a waste of memory. I mean is it extremely likely that in the "real" world of physics that you foul up terribly because you forgot about the Squeeze Theorem? When will I actually have to compute or work with a limit directly?
|Mar5-12, 11:38 PM||#2|
So in calculus what you should do is just accept whatever it says in the book about limits and their properties; and then make sure you learn all the techniques of differentiation and integration.
As an engineer or a physicist you won't often have any need to care about the logical foundation of the real numbers and limits. And if you do you can always ask here :-)
The answer to when you will need to really understand the meaning of limits is not till you take a course in real analysis.
I do realize that there are some modern textbooks in calculus that take a more rigorous approach. Whether this is a good idea pedagogically for physics and engineering students, I can't say. But if you're in a traditional "bring down the exponent and subtract 1" calculus class, you should concentrate on knowing how to apply the chain rule, not necessarily prove it; unless your class is proof-based.
I hope I haven't said anything too inflammatory. The question of what to tell freshmen about limits is one that generates a lot of opinions. Do whatever your teacher says to do, that's always the best advice.
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