# Questions tagged [limits]

Questions on the evaluation and properties of limits in the sense of analysis and related fields. For limits in the sense of category theory, use (limits-colimits) instead.

41,756 questions
Filter by
Sorted by
Tagged with
31 views

### Maximise area of rectangle with fixed perimeter

I've got a problem where a rectangle's area must be maximised given a fixed perimeter of $60$m. Assuming a length of $x$ and height of $y$ I wrote an equation $y = 30x - x^2$ which i differentiated, ...
19 views

• 799
1 vote
54 views

### There is $R > 0$ such that there are no holomorphic functions $f: \{z: |z| < R \} \to \mathbb C \backslash \{0,1\}$ with $f(0) = a$ and $f'(0)= b$

Suppose that $a,b \in \mathbb C \backslash \{0\}, a \neq 1$. Prove that there is some $R > 0$ such that there are no holomorphic functions $f: \{z: |z| < R \} \to \mathbb C \backslash \{0,1\}$ ...
• 2,981
1 vote
39 views

### What is the concept of a limit in simple terms

I'm 14 just starting high school and I got interested in all the new math so I took a little dive into calculus but I can't wrap my head around the concept of a limit and for which uses it is applied ...
27 views

### Limit Derivation and Proof (Rigorous ε-δ Proof) [closed]

Could someone, please, provide me with a rigorous ε-δ proof for the following: $$\lim_{x \rightarrow 4^{-}}\frac{\sqrt{x}-1}{(x^{2}-16)^{5}}=-\infty$$ ?
1 vote
65 views

### Prove limit of a function that goes from $\mathbb{R}^{2} \rightarrow \mathbb{R}$

I need to find $\lim_{(x,y) \to (0,0)} f(x,y)$ where $f(x,y) = \frac{\sqrt{(1 + 4x)(1 + 6y)}-1}{2x + 3y}$. I have already computed such limit by fixing $y = 0$, and it turned out it went to 1, and ...
• 159
58 views

### Does this limit exist and if it does what is it? [closed]

I was playing around with a few values of $k$ between $0$ and $1$ on Wolfram alpha and found that for all of them $$\sum_{n= 1}^\infty \frac{\sin(n)}{n^k}$$ converges and I was wondering about the ...
22 views

66 views

77 views

### a function defined by inferior limit related to Borel measure is Borel measurable

Question: $\mu$ is a Borel measure on $\mathbb R$. Define $f:\mathbb{R\to \bar R},f(x)={\operatorname{lim inf}}_{r\to 0}{{\mu((x-r,x+r))}\over{r}}$. Prove that $f$ is (extended) Borel measurable. I ...
• 589
94 views

### Are there other, non-probabilistic ways to calculate: $\lim_{n\to\infty}\frac{1}{n}\ln\sum_{m>n\alpha}\frac{(n\lambda)^m}{m!}$?

In section five of this nice exposition of moment generating functions, we prove the following theorem: Take an i.i.d sequence of random variables $(X_n:\Omega\to\Bbb R)_{n\in\Bbb N}$ whose common ...
• 22k
51 views

### Find the limit $\lim_{n→∞} \frac{n+k}{n^2} \sum_{k=1}^n (\ln(n+k) - \ln(n))$ [closed]

My exercise is: $$\lim_{n→∞} \sum_{k=1}^n \frac{n+k}{n^2}(\ln(n+k) - \ln(n))$$ I don't know how to solve it? Can you give me advices?