![]() is there a smallest particule of time, matter, etc. there were some discussion about the possible existence of an "infinitesimal", i.e. This is how we would answer, e.g., the commonplace question "how fast was he going at time $x$?".Ī bit of History: Already in 500 b.c. That is to say, we want to measure a quantity in an instant, and we define this "instant" by a limit, i.e., as an approach towards some infinitesimal time. Now this is dependent on the concept of the limit. We'd like to be able to measure instantaneous speed, which requires the notion of an instantaneous value. To apply this notion to physics (yes, I'm moving away from math now), it is possible to apply a continuous analysis to motion. So more generally, the limit helps us move from the study of discrete quantity to continuous quantity, and that is of prime importance in Calculus, and applications of Calculus. Those ideas are not trivial, and it is hard to place them in a rigorous context without the notion of the limit. Limits are super-important in that they serve as the basis for the definitions of the 'derivative' and 'integral', the two fundamental structures in Calculus! In that context, limits help us understand what it means to "get arbitrarily close to a point", or "go to infinity". It is hard for me to stray from the confines of mathematics to the 'real world', so let me give you this "example": ![]() What is a simple example of a limit in the real world? 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. So limits are important what I've just described is trivial. 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). To me it looks like an elementary-algebra problem ($2p = 10$). 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. To me that sounds like something that might be better described as a 'target'. In other words, $f(x)$ becomes closer and closer to $L$ Theįunction has a limit $L$ at an input $p$ if $f(x)$ is "close" to $L$ whenever Informally, a function f assigns an output $f(x)$ to every input $x$. This morning, I read Wikipedia's informal definition of a limit:
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