Wikipedia:Reference desk/Archives/Mathematics/2024 June 25

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June 25

Defining a function involving a limit

If you are defining g(x), why is it written

$\lim _{x\to \pi }{\sqrt {x^{3}f(x)}}=g(x)\,$,

and not

$g(x)=\lim _{x\to \pi }{\sqrt {x^{3}f(x)}}\,$,

or are both ways acceptable?

I mean you usually don't say ${\sqrt {x^{3}f(x)}}=g(x)\,$, you say $g(x)={\sqrt {x^{3}f(x)}}\,$, right? (Yes, g(x) is purely nonsensical in meaning, for demonstrative purposes only :)

173.14.155.129 (talk) 23:20, 25 June 2024 (UTC)

It is definitely more conventional to place the entity being defined in the lhs of the equation, but as long as it is clear which is the definiendum and which the definiens, there is nothing logically wrong with the swapped version – it is merely highly unusual and therefore may put the reader on the wrong foot. This is independent of whether the definiens involves a limit. By the way, the value of the expression $\lim _{x\to \pi }{\sqrt {x^{3}f(x)}},$ if defined, does not depend on $x.$ If function $f$ is continuous at $\pi ,$ it is equal to ${\sqrt {\pi ^{3}f(\pi )}}.$

 

When asked to determine

$\lim _{~\epsilon \to 0^{+}}(1+\epsilon x)^{\frac {1}{\epsilon }},$

I'd present the answer in the form

$\lim _{~\epsilon \to 0^{+}}(1+\epsilon x)^{\frac {1}{\epsilon }}=e^{x},$

but when asked to define the exponential function, I might give

$\exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}$

as one of several definitions, and then I'd present it in this order.  --Lambiam 00:02, 26 June 2024 (UTC)

Like I said, that was just a bogus example (I'm actually thinking of a segmemt that changes as it grows, so that the midpoint value of the whole is different than the midpoint value calculated "locally", with an infinitesimal length), but you answered the question.

173.14.155.129 (talk) 01:03, 26 June 2024 (UTC)

Quick side note that equality in the sense of definition - as opposed to equality in the sense of identity - is often denoted with a colons-equal := (as a single character, ≔), in which case I believe the function $g(x)$ being defined would have to go on the left. GalacticShoe (talk) 05:41, 26 June 2024 (UTC)

What is nonsensical here is (IMHO) using a limit as $x\to \pi$ to define $g(x)$. I mean, assuming the limit of an expression with a variable $x$ used in the actual problem exists, that limit no longer depends on $x$. As such the expression defines $g(x)$ as a constant function... Could you, please, quote the relevant part of the actual problem you're solving? --CiaPan (talk) 12:45, 26 June 2024 (UTC)

This was already noted above, but, as the poster themself wrote, the definition was "purely nonsensical in meaning, for demonstrative purposes only". The problem for which they were seeking a solution was, why, when the definiens involves a limit, does the definiendum become the rhs of the defining equation? The answer is that it actually dpes not change position but stays as the lhs.  --Lambiam 15:04, 26 June 2024 (UTC)

Then something like 'defining Y with X by X=Y' would be more clear, not obscuring the core problem with a meaningless babbling... --CiaPan (talk) 13:04, 27 June 2024 (UTC)

Tha's why I wrote, "as long as it is clear which is the definiendum and which the definiens, there is nothing logically wrong with the swapped version". Apparently – or so I guess – the poster saw an equation of the form $\lim _{z\to x}f(z)=g(x)$ and incorrectly assumed the intention of this equation was to define $g.$  --Lambiam 16:28, 27 June 2024 (UTC)

I would agree it is okay to express the limit on the right side, so that (using a more classical example),

$f'(x)={\frac {df(x)}{dx}}=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

I suppose you could even say $d=\lim _{h\to 0}h$, or better still $d=\lim _{\Delta \to 0}\Delta$ (?)

On a related notation question (where C is a constant),

if h(x) = h(C,x) = f(C)g(x) does

$h'(x)=h'(C,x)={\frac {dh(x)}{dx}}={\frac {\partial h(C,x)}{\partial x}}=\lim _{D\to 0}{\frac {h(C,x+D)-h(C,x)}{D}}=??{\frac {dh(C,x)}{dx}}$?

Since C is a constant, doesn't

$\int _{a}^{b}h'(x)dx=\int _{a}^{b}h'(C,x)\partial x=\int _{a}^{b}h'(C,x)dx$?

If not, what does

$\int _{a}^{b}h'(C,x)dx$ equal?   2601:188:CB81:CBC0:55BF:ADB0:FA15:C41D (talk) 23:48, 29 June 2024 (UTC)

Mathematical notation is not always as unambiguous as it is sometimes believed to be, but one should draw the limit somewhere. The conventional meaning of $\lim _{h\to 0}h$ is that $\lim _{h\to 0}h=0,$ so in your shorthand notation you appear to be defining $d=0.$  --Lambiam 07:46, 30 June 2024 (UTC)

As to the second question, the issue is made more complicated than necessary by overloading the function name $h.$ Let it stand solely for a function of two variables. Then, given a constant $C,$ we can define $k(x)=h(C,x).$ Taking the derivative with respect to $x,$ we can simply write,

$k'(x)={\frac {d}{dx}}h(C,x).$

There is no need in this case to use the notation for partial derivatives. The notation $h'(C,x)$ has no clear meaning and should not be used, so the final question is moot.  --Lambiam 12:38, 30 June 2024 (UTC)