The Real Importance of Limits in Calculus

[nDever]

The "importance" of Limits

Hey,

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?

 

[SteveL27]

Hey,

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?

Limits are not very important in calculus. They are important when you study Real Analysis, or "Calculus made legit." The problem is that in calculus you can't really define a limit properly, because you don't have a construction of the real numbers that let's you prove the least upper bound property, also known as the completeness property of the reals. The completeness property says there are no "holes" in the reals. It's what you need to rigorously prove the Intermediate Value Theorem, for example.

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.