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Apr
25
comment Why doesn't L'Hospital's rule work for this limit?
The rule (for this particular indeterminate form) says: If $f(x)\to \infty$ and $g(x)\to \infty$ and $\lim\frac{f'(x)}{g'(x)}$ exists, then $\lim\frac{f'x)}{g(x)}$ exists and equals $\lim\frac{f'(x)}{g'(x)}$.
Apr
25
comment The sum of two numbers is 5/9…
I read the last instruction as asking for $(0.4 x)\cdot (0.4y)$
Apr
25
answered Does there exist a complex function which is differentiable at one point and nowhere else continuous?
Apr
25
comment Has the polynomial distinct roots? How can I prove it?
@drxy I found my counterexample by just playing around with $\gcd(f,f')$, motivated by mathguy's solution for real roots. I'm sure that playing around will also allow finding counterexamples (regarding complex roots) for higher $p$ values one by one - but the handling becomes impractical.
Apr
25
comment Prisoners and hats variation
Do they guess only once? OR is the game repeted (with the same hats) if nobody dared to guess?
Apr
25
comment $\sup\limits_{t>0}[\frac{g(t)}{\sup\limits_{t<u<\infty}g(u)}]\leq 1$
To fleshen out the last sentence: Replace $2^{-n}$ with $\frac1{n!}$
Apr
25
answered How can I show that $G$ is non abelian of order 20?
Apr
25
answered You are given two points and a circle. Construct a circle passing through the given two points and tangent to the given circle.
Apr
25
comment $\frac{0}{0}$ Indeterminate
The notion "indeterminate form" relates to the fact that $a_n\to 0$, $b_n\to 0$ does not tell us much about the limit of $\frac{a_n}{b_n}$. Howver, $\frac 00$ is also "undefined" for the simple reason that it is not immediately defined per the usual definition of $\frac ab$ as the unique solution (if it exists) of $b\cdot x=a$. Lacking uniqueness in the case $a=b=0$, this definition does not cover $\frac 00$. -- In contrast to this, $0^0$ is an "indeterminate form" (in the unkown-limit-sense above), but it is defined (contrary to somewhat popular belief)
Apr
25
revised The $\cos(\alpha-\beta)$ formula always need $\alpha > \beta$ or not?
deleted 1 character in body
Apr
25
comment The $\cos(\alpha-\beta)$ formula always need $\alpha > \beta$ or not?
@user3270418 Where do I show the sum formula? I read $\cos(\alpha-\beta)$ in my post.
Apr
25
answered Significance of derivative in finding square free decomposition
Apr
25
answered Deterministic finite automaton parity bit question
Apr
25
comment The $\cos(\alpha-\beta)$ formula always need $\alpha > \beta$ or not?
Are you sure your argument leads to a "No"? - Oh, you seemingly answered the title question, which is the opposite of the body question ...
Apr
25
comment The $\cos(\alpha-\beta)$ formula always need $\alpha > \beta$ or not?
@user3270418 No, I mean what I wrote. For $\alpha=\beta=60^\circ$, we have $\cos(60^\circ)^2+\sin(60^\circ)^2=(1/2)^2+(\sqrt 3/2)^2=1/4+3/4=1$
Apr
25
answered The $\cos(\alpha-\beta)$ formula always need $\alpha > \beta$ or not?
Apr
24
answered Show that there is a step function $g$ over $[a,b]$
Apr
24
comment Is there a way to write an infinite set that contains only irrational numbers without integer multiples?
In the same spirit as the last, $\{\,q\sqrt 2\mid q\in \Bbb Q\cap [1,2)\,\}$
Apr
24
comment The maximum of the absolute value of a real-valued function
You could simply use that $\sup\bigcup_{i\in I}A_i =\sup_{i\in I}sup A_i$
Apr
24
answered Probability of seeing a headlight getting switched on