Intereting Posts

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How to solve probability when sample space is infinite?
Is it possible to practice mental math too often?
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
Find the expected number of two consecutive 1's in a random binary string
How can I define a topology on the empty set?
Show that $\lim_{n\rightarrow \infty} \sqrt{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$
Expected revenue obtained by the Vickery auction with reserve price $1/2$
Galois Group of $x^{4}+7$
Fredholm operator norm
Limit. $\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$
Irreducible polynomial over Dedekind domain remains irreducible in field of fractions
There is a well ordering of the class of all finite sequences of ordinals
Proof Verification : Prove -(-a)=a using only ordered field axioms
What is the intuitive meaning of the scalar curvature R?

In the text I’m using (Spivak’s *Calculus*, 4E), it is established (problem 5.39(iii)) that $$\mathop{\lim}\limits_{x \to \infty}\left({x\space\sin^{2} x}\right)$$ “does not exist”. It is also established (5.39(c)) that [A] if $\mathop{\lim}\limits_{x \to \infty}f(x)$ exists, but $\mathop{\lim}\limits_{x \to \infty}g(x)$ does not, then $\mathop{\lim}\limits_{x \to \infty}\left[f(x)+g(x))\right]$ cannot exist.

But the text also establishes (5.39(ii)) that $$\mathop{\lim}\limits_{x \to \infty}\left({x+x\space\sin^{2}x}\right)=\infty.$$

which seems to be a contradiction of the just established property of limits, with $f(x)=x$ and $g(x)=x\space\sin^{2} x$.

- Toward “integrals of rational functions along an algebraic curve”
- The magnitude of the difference between the integral and the Riemann sums for continuous functions
- On $\big(\tfrac{1+\sqrt{5}}{2}\big)^{12}=\small 161+72\sqrt{5}$ and $\int_{-1}^1\frac{dx}{\left(1-x^2\right)^{\small3/4} \sqrt{161+72\sqrt{5}\,x}}$
- Prove that $\int _0^1x^a\left(1-x\right)^bdx$ = $\int _0^1x^b\left(1-x\right)^adx$, where $a,b\in \mathbb{R}$
- Integral equation
- Optimization with cylinder

Is there something important going on here with regard to limits that “equal” infinity?

- What is the limit of this sequence involving logs of binomials?
- Evaluating the sum of geometric series
- Finding the slope of the tangent line to $\frac{8}{\sqrt{4+3x}}$ at $(4,2)$
- Finding the limit of $\left(\sum\limits_{k=2}^n\frac1{k\log k}\right)-\left(\log \log n\right)$
- The length of toilet roll
- Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$
- How to calculate a limit of this function for checking integral convergence?
- How do I prove that a function with a finite number of discontinuities is Riemann integrable over some interval?
- Check my workings: Show that $\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$
- Show that the function is discontinuous in $\mathbb{R}$

This does not contradict the established property of limits. If you look closely, those are established only when the limits exist! This provides a loophole for this case.

As for this limit, it goes back to what we mean when we write $\lim_{x \rightarrow \infty} f(x) = \infty$. In particular, this means for any large $M$ I choose, you can find some large enough $a$ such that $f(x) > M$ for all $x > a$.

For $x\sin^2(x)$, there are infinitely many positive values for $x$ that make $x\sin^2(x)$ zero. In particular, there exists some $M$ ($M = 0$ works) such that for all $a$, there exists some $b > a$ with $b\sin^2(b) = 0$. Convince yourself that this is the negation of the above definition, so that we’ve genuinely shown that $\lim_{x \rightarrow \infty} x\sin^2(x) \neq \infty$. Using a similar argument, I bet you can show that $\lim_{x \rightarrow \infty} x\sin^2(x)$ does not equal anything else, either. You’re right above when you say “it never settles on a single value”. However, you’re incorrect when you say it is “sometimes $\infty$”; this makes no sense to say.

It should not be difficult to use the definition to show $\lim_{x \rightarrow \infty}(x + x\sin^2x) = \infty$. In particular, choose some arbitrary $M$ and find some value $a$ (it will depend on $M$!) such that $x + x\sin^2x > M$ for all $x > a$.

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- Evaluating limits at positive and negative infinity
- $F,G \in \text{End} (V)$ share the same eigenvalues for $F \circ G$ and $G \circ F$
- Initial Value Problem
- Local vs. global in the definition of a sheaf
- On the limits of weakly convergent subsequences
- using the same symbol for dependent variable and function?
- $\dim C(AB)=\dim C(B)-\dim(\operatorname{Null}(A)\cap C(B))$
- Let $R=M_n(D)$, $D$ is a division ring. Prove that every $R-$simple module is isomorphic to each other.
- A coin is ﬂipped 8 times: number of various outcomes