What is a simple example of a limit in the real world?

Question

This morning, I read Wikipedia's informal definition of a limit: > Informally, a function f assigns an output $f(x)$ to every input $x$. The > function has a limit $L$ at an input $p$ if $f(x)$ is "close" to $L$ whenever > $x$ is "close" to $p$. In other words, $f(x)$ becomes closer and closer to $L$ > as $x$ moves closer and closer to $p$. To me that sounds like something that might be better described as a 'target'. If I take a simple function, say one that only multiplies the input by $2$; and if my limit is $10$ at an input $5$: then I've described something that seems to match the elements contained in Wikipedia's definition. I don't believe that that's right. To me it looks like an elementary-algebra problem ($2p = 10$). To make it more calculusy, I could graph the function's output when I use inputs other than $p$, but that really wouldn't give me anything but an illustration of the fact that one's answer moves farther from the right answer as it becomes more wrong (go figure). So limits are important; what I've just described is trivial. I do not understand them. I know calculus is often used for solving real-world challenges, and that limits are an important element of calculus, so I assume there must be some simple real-world examples of what it is that limits describe. What is a simple example of a limit in the real world? Thank you -Hal.

Answer 1

This may be too simplistic an example for you, but the best real world example of a limit is the speedometer in your car! The speedometer measures instantaneous velocity, i.e. the velocity right now.. How would you define that? Well, calculus defines it as the limit of the change in distance traveled in a time period as the time period becomes smaller.

Second example: you know how to define the area of a rectangle (a garden plot, a sidewalk)...as the product of the length and width. From that it is a fairly easy step to define the area of a figure "composed" of all straight lines. So you can figure out how much paint to buy to paint any such figure. Now you need to know how much paint to buy to paint a circular water tower. At least one way to define the area of non-rectilinear figures involves the notion of a limit, the limit of the areas on enclosing and enclosed rectangles as one of the dimensions of the rectangle becomes smaller.

Each of these examples involves the fundamental notions of calculus, the derivative and the integral.

And BTW, no "informal" description of a limit will do the job...when you get to calculus you MUST memorize (yes, memorize) the definition of a limit...because if you don't get the definition of a limit correct, your study of calculus will be hopeless!

Answer 2

I have Thomas's Calculus 14th ed. It does a very good job of describing limits (the best I have seen).

For your example, you said f(x) = 2x.

You claimed (correctly) that the lim f(p) as p → 5 is 10. You are right. 10 is the target that the outputs get closer to as p gets close to 5. It is easy to find limits for well-behaved, continuous functions. Often, all that is needed is a little plug-n-chug. Unfortunately, most functions are not well-behaved and some are even pathological. (See https://en.wikipedia.org/wiki/Weierstrass_function for an example of a pathological function.)

Practical Applications:

1.) Rates of Change

2.) Continuously Compounded Interest

3.) Defining an irrational number (like π or e) Happy pi day, by the way!

4.) Pretty much any result from calculus (derivatives and integrals all depend on limits)

5.) Limits at infinity are useful for describing the end behavior of a function