My progress: Using the sequential criterion for limits, I constructed two sequences $(x_n), (y_n)$ with $\lim(x_n)=\lim(y_n)=0$, such that $\lim(f(x_n))\neq \lim(f(y_n))$, where $f(x)=\sin\frac 1 x$. So, $\lim_{x \rightarrow 0} \sin (\frac{1}{x})$ does not exist. I also showed separately in the same way that $\lim_{x \rightarrow 0} \mathrm{sgn} (x)$ does not exist. I know that $$\lim_{x \rightarrow […]

Let $\{X_n\}_{n=1}^\infty$ be a convergence sequence such that $X_n \geq 0$ and $k \in \mathbb{N}$. Then $$ \lim_{n \to \infty} \sqrt[k]{X_n} = \sqrt[k]{\lim_{n \to \infty} X_n}. $$ Can someone help me figure out how to prove this?

Let $f: [0,\infty)\to \Bbb R$ be a positive,decreasing monotonic function. Prove the following statement for every a>0 providing the integral on the right side converges. First I managed to prove that the series on the left size converges using the integral test, but now I’m having a hard time proving that the equality above is […]

Let $u(x,y)\to l$ as $(x,y)\to z_0=(x_0,y_0).$ How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$ Choose $\epsilon>0.$ Then $\exists~\delta>0$ such that $|u(x,y)-l|<\epsilon~\forall~(x,y)\in (B(z_0,\delta)-\{(x_0,y_0\})\cap D$ ($D$ being the domain of $u$). Of course the domain of $u(x,y_0)$ is $D_1=\{x:(x,y_0)\in D\}$ Thus for all $x\in((x_0-\delta,x_0+\delta)-\{x_0\})\cap D_1, |u(x,y_0)-l|<\epsilon.$ This far is […]

A part of the proof says that if $n\le N$, then, the sequence $x_n \le \max\{|x_1|,|x_2|,….,|x_{N-1}|\}$. I’m not capturing the intuition of the above. Even more perplexing is, if it the case where the sequence is monotonic decreasing, then for every $n\le N$, it is obvious that $x_n$ will not be $\le \max\{|x_1|,|x_2|,….,|x_{N-1}|\}$. I greatly […]

I am reading a text, where the neutral curve as a result of linear stability analysis of a delayed differential equation is given. $\delta$ and $\alpha$ are parameters of this model. The curve is defined as $$ \delta=\arccos\left(\frac{3\alpha-2}{\alpha}\right)\frac{1}{\sigma_i(\alpha)},$$ where $$\sigma_i(\alpha)=\sqrt{\alpha^2-(2-3\alpha)^2}.$$ Furthermore, I know that $\delta\gt 0$ and $0\lt\alpha\leq 1$. I do understand how these equations […]

I have the following question: Let $(x_n)$ a sequence in $X$ and $x\in X$ such that for all $F\in X’$ (the dual space of the vector space X) we have that $(F(x_n))$ converge to $F(x)$ (that is: the sequence converge weakly on $X$). Let $F:X\longrightarrow\mathbb{R}$ a continuous function. Is it true that $\quad\displaystyle\liminf_{n\to\infty}|f(x_n)|\;{\color{red}\geq}\;{\color{red}|}f(x){\color{red}|}$? Recall that: […]

This question already has an answer here: How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? 17 answers What does $\sum_{k=0}^\infty \frac{k}{2^k}$ converge to? 2 answers

I define $\exp: \mathbb C \to \mathbb C$ as $z \mapsto \sum \limits_ {k=0}^{\infty}\frac{z^k}{k!}$. I would like to show that $\lim \limits_{n\to\infty}(1+\frac{z}{n})^n = \exp(z)$. I have a proof for the case $z \in \mathbb R$, but the proof assumes that $\lim \limits_ {n\to\infty}(1+\frac{z}{n})^n$ exists, which is easy to see if $z \in \mathbb R$, but […]

Suppose $(a_n)$ and $(b_n)$ are sequences where $b_n$ is increasing and approaching positive infinity. Assume that $\lim_{n\to \infty}$ $\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=L$, where $L$ is a real number. Prove that $\lim_{n\to \infty}$ $\frac{a_n}{b_n}=L$.

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