Intereting Posts

Show that $\frac{(3^{77}-1)}{2}$ is odd and composite
Continuous image of a locally connected space which is not locally connected
Given any base for a second countable space, is every open set the countable union of basic open sets?
Prove that, there exists no continuous function $f:\mathbb R\rightarrow\mathbb R$ with $f=\chi_{}$ almost everywhere.
If $q^k n^2$ is an odd perfect number with Euler prime $q$, which of the following relationships between $q^2$ and $n$ hold?
Derivation of the Dirac-Delta function property: $\delta(bt)=\frac{\delta(t)}{\mid b \mid}$
Why are powers of coprime ideals are coprime?
Cartesian product of two CW-complexes
nonlinear transform of Gaussian random variable that preserves Gaussianity
To confirm the Novikov's condition
Showing diffeomorphism between $S^1 \subset \mathbb{R}^2$ and $\mathbb{RP}^1$
pullback of rational normal curve under Segre map
Prove $p_n(x) \rightarrow \sqrt{x}$ uniformly as $n \rightarrow \infty$
Cartesian product of compact sets is compact
convergence of $\sum \limits_{n=1}^{\infty }\bigl\{ \frac {1\cdot3 \cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \frac {4n+3} {2n+2}\bigr\} ^{2}$

Why $\displaystyle\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim_{t\to 0}\frac{\sin t}{t}$( and hence equals to $1$)?

Any rigorous reason? (i.e. not just say by letting $t=x^2+y^2$.)

- Find $\lim (a_{n+1}^\alpha-a_n^\alpha)$
- The distance between two sets inside euclidean space
- Where's the error in this $2=1$ fake proof?
- Hölder continuous but not differentiable function
- Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional
- measurable functions and existence decreasing function

- Show that every large integer has a large prime-power factor
- Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$
- show $\exists\ m\in\mathbb{N} \text{ such that: } \quad a+\dfrac{m}{2^n}\geq b$
- Constructing Continuous functions at given points
- Integrate $\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$
- partial derivatives continuous $\implies$ differentiability in Euclidean space
- Show that integral involving $\frac {x^{a}-x^{b}}{(1+x^{a})(1+x^{b})}$ is actually zero for every $(a,b)$
- How to prove that a set R\Z is open
- Proof that rational sequence converges to irrational number
- Doubling measure is absolutely continuous with respect to Lebesgue

Here’s a point of view I like. The fact that $$\lim_{t\to0}\frac{\sin t}{t}=1$$ means precisely that the function $\phi:\mathbb R\to\mathbb R$ defined by $$\phi(t)=\begin{cases}\frac{\sin t}t;&t\neq0,\\1;&t=0,\end{cases}$$ is continuous. Therefore, we have $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim_{(x,y)\to(0,0)}\phi(x^2+y^2)=\phi(0)=1.$$ The first equality holds because the definition of limit doesn’t involve the value of the function at $(0,0)$ and the second equality holds because $\phi(x^2+y^2)$ is continuous (since it is the composition of two continuous functions).

Using $x = t \cos \alpha $ and $y = t \sin \alpha$, then:

$$ \frac{ \sin (x^2 + y^2 ) }{x^2+y^2} = \frac{ \sin (t^2(\sin^2 \alpha + \cos^2 \alpha)) }{t^2(\sin^2 \alpha + \cos^2 \alpha)} = \frac{ \sin (t^2) }{t^2}$$

.

Note as $x,y \to 0,0$, then $t \to 0 $.

- Question on geometrically reduced, geometrically connected.
- Understanding the relationship of the $L^1$ norm to the total variation distance of probability measures, and the variance bound on it
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- Proving an Entire Function is a Polynomial
- Sum of consecutive square roots inside a square root
- True or false: {{∅}} ⊂ {∅,{∅}}
- How can I show that a sequence of regular polygons with $n$ sides becomes more and more like a circle as $n \to \infty$?
- How to find $x$ when $2^{x}+3^{x}=6$?
- Uses of quadratic reciprocity theorem
- Proving $\sqrt{ab} = \sqrt a\sqrt b$
- Tower property of conditional expectation
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- integral to infinity + imaginary constant
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- A is Mn×n(C) with rank r and m(t) is the minimal polynomial of A. Prove deg $m(t) \leq r+1$