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March 6

Limits of partial functions

At first glance it looks like

\lim _{x\to 0}x{\sqrt {\sin {\frac {1}{x}}}}=0

since whatever is under the square root is bounded and x→0. But both of my undergraduate analysis texts say no because the domain of the function does not include a neighborhood of 0. (One-sided limits are defined separately in these sources.) If

D:=\{x:x\neq 0\,{\rm {{and}\,\sin {\frac {1}{x}}\geq 0\}\cup \{0\}}}

and

f(x):={\begin{cases}x{\sqrt {\sin {\frac {1}{x}}}}&x\neq 0\\0&x=0\end{cases}}

for x∈D, then we have the odd situation where f is continuous on D, f(0)=0, but $\lim _{x\to 0}f(x)$ is undefined. Are there alternative definitions for limit which avoid this issue? Note that the WP article Limit of a function, and presumably most freshman calc texts gloss over this issue since they use the expression | f(x) − L | < ε without explicitly requiring that it be defined. --RDBury (talk) 07:59, 6 March 2018 (UTC)[reply]

In my view,

\lim _{x\to 0}f(x)=0

here, because for all x in D with

|x|<\delta =\epsilon

,

|f(x)-0|<\epsilon

. D is a metric space (as a subset of the real line), and I would use the definition of convergence in that metric space. —Kusma (t·c) 12:03, 6 March 2018 (UTC)[reply]

Kusma is correct, per the usual definition of limit in R. Indeed, f(x)=x*sin(1/x) (with f(0)=0) is the textbook function to show students (well, at least that's the one I was shown) that a function can be continuous everywhere yet not have any derivative left or right (see left derivative). Tigraan^{Click here to contact me} 20:10, 8 March 2018 (UTC)[reply]

See the section More general subsets at Limit of a function#More general subsets, IMHO it precisely addresses your problem. --CiaPan (talk) 12:41, 6 March 2018 (UTC)[reply]
Forgot to ping (again!): @RDBury:. --CiaPan (talk) 12:42, 6 March 2018 (UTC)[reply]

Thanks; I missed that. There are no sources listed for the section though so it's hard to tell if that notation is generally accepted. @Kusma:, the more general topological approach does include the 'More general subsets' definition as a special case. I can see why it's not used as an initial definition because of the limit point requirement, needed to guarantee the limit is well-defined. I'll check my texts again to see if they generalize their definition later on. --RDBury (talk) 15:22, 6 March 2018 (UTC)[reply]

Principle

What's the principle called when you choose one individual out of a sample and infer that it will be representative of the entire sample? 93.140.132.12 (talk) 08:24, 6 March 2018 (UTC)[reply]

Inductive reasoning. Bo Jacoby (talk) 12:54, 6 March 2018 (UTC).[reply]

Or cherrypicking. --JBL (talk) 21:55, 6 March 2018 (UTC)[reply]

Or in general, faulty generalization. See also List of fallacies#Faulty generalizations. --69.159.62.113 (talk) 19:22, 7 March 2018 (UTC)[reply]

Asymptotic equivalent for product of primes

Is there an asymptotic equivalent, or a lower bound, for the product of the n first prime numbers? Said less formally, for the sequence $\Pi (1)=2,\Pi (2)=2*3,\Pi (3)=2*3*5,...$ , what is the rough value of $\Pi (n)$ ?

Context in order to avoid the XY problem: I am toying around with modular arithmetic in Python to store (large) integers as the set of their residuals modulo the n first primes; the Chinese remainder theorem ensures that each such representation is matched to a unique integer between 0 and $\Pi (n)-1$ . From an upper threshold M on the largest integer I will manipulate, I want to find the number of primes I need, i.e. find n such that $\Pi (n)>M$ , but without taking too large a n either. It is okay to make a few iterations if the initial guess is not exactly right, but I have no clue as to what the initial guess should be right now. Tigraan^{Click here to contact me} 17:25, 6 March 2018 (UTC)[reply]

These numbers are called Primorial numbers and there are asymptotic formulas in the article. Btw, if you're using the CRT to store integers it might be more efficient to use prime powers. For example if you're going with 4 values between 0 and 10 then using 8, 9, 5, 7 as moduli you can store numbers up to 2520 instead of just 210 which you get using primes. --RDBury (talk) 18:05, 6 March 2018 (UTC)[reply]

Thanks for the pointer. As for the efficiency, well it is not a big deal since it is a toy project, but (as I am sure you know) modular arithmetic is easier with a prime modulus (for instance exponentiation to a large power is trivial with Fermat's little theorem). Tigraan^{Click here to contact me} 13:23, 7 March 2018 (UTC)[reply]

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