Articles of limits

Rate of convergence of Cesàro means

For a sequence $a_n = O(n^{-1/2})$ as $n\to\infty$, consider the corresponding Cesàro means $b_n = \frac{1}{n} \sum_{j=1}^n a_j$. Is it possible to derive the rate of convergence for the sequence $b_n$? What about the general case $a_n = O(c_n)$?

What is the limit $\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$

If $\vartheta_{2}(q)$ is jacobi’s theta function, what is the limit $$\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$$ for the nome $q$. I would like to know whether the limit exists or not. If it does, please let me know and provide it’s evaluation

$\lim_{x\to\infty}\int_x^{x+1}f(y)dy=0$ implies $\lim_{x\to\infty}\frac{\int_0^{x}f(y)dy}{x}=0$

Let $f$ be a non negative continuous function on $[0,\infty)$ such that $$\lim_{x\to\infty}\int_x^{x+1}f(y)dy=0.$$ How do we prove that $$\lim_{x\to\infty}\frac{\int_0^{x}f(y)dy}{x}=0.$$ If we see this question using the primitive function, do we have the following result for a continuous function $F$: $$\lim_{x\to\infty}F(x+1)-F(x)=0$$ implies that $$\lim_{x\to\infty}\frac{F(x)}{x}=0.$$

Is $e^x$ in the span of $\{1,x, x^2,…\}$ in the vector space $C$ on $$?

T. Gunn’s answer to a question about linear combinations being restricted to finite sums asserted that For example, you would not count the “infinite” linear combination $$\exp(x) = \sum_{n = 0}^\infty \frac{x^n}{n!} $$ to be in the span of $\{1,x,x^2,\dots\}$ in the vector space $C[0,1]$ of continuous functions on $[0, 1]$. After some brief reading, […]

solve $\lim_{x\rightarrow \infty}\frac{x^3}{e^x}$ using L'Hospitals rule

I am having problems with this question. I know the answer is 0 but I keep getting infinity over infinity. I am using L’Hospitals rule. Any help would be much appreciated 🙂 edit* I only used L’Hospitals rule once. edit*** Now it works when I use it 3 times 🙂

Find the limit $\lim_{x\to0}\frac{\arcsin x -x}{x^2}$

How can I find $\lim_{x\to0}\frac{\arcsin x -x}{x^2}$? I’ve tried using the Lhopital rule and it got me here: $\lim_{x\to0}\frac{\arcsin x -x}{x^2} = \lim_{x\to0}\frac{\frac{1}{\sqrt{1-x^2}}-1}{2x} = \lim_{x\to0}\frac{x}{2\sqrt{1-x^2}\cdot (1+\sqrt{1-x^2})}$ This doesn’t make life much easier, unless I could say that $\frac{x}{2\sqrt{1-x^2}\cdot (1+\sqrt{1-x^2})}$ is continuous at $x=0$.. Is there a better way to approach this?

When is a balance assumption consistent?

From Asymptotic analysis and perturbation theory by Paulsen: Find the behavior of the function defined implicitly by $$x^2+xy-y^3=0$$ as $x\to\infty$. […] The final case to try is to assume that $xy$ is the smallest term. Then $x^2 ∼ y^3$, which tells us that $y ∼ x^{2/3}$. To check to see if this is consistent, we […]

How to prove that $\lim_{x\to 0}f(x)=L$ is equivalent to $\lim_{x\to 0}f(x^3)=L$ rigorously

I’ve tried to prove this by epsilon-delta, but it didn’t go well…

Limit of Root: Shifted

It is a well known fact that: $$\lim_{N\to\infty}\sqrt[n]{n}=1$$ But what about the shifted one up: $$\lim_{N\to\infty}\sqrt[n+1]{n}=1$$ And what about the shifted one down: $$\lim_{N\to\infty}\sqrt[n]{n+1}=1$$

Another simple/conceptual limit question

Further to my previous post Very simple limits question to clarify my understanding , here’s a related question. Let $f(x)=\sqrt x,x\geq 0$. What is the limit of $f$ as $x$ tends to $0$? I think the answer is $0$, but my textbook claims that the limit doesn’t exist because $f$ is not defined on any […]