# Tagged Questions

Questions on the evaluation of limits.

122 views

### $\lim_{x\to c}f'(x)=L$ implies $f'(c)=L$

Let $f:[a,b]\to\mathbb{R}$ be differentiable on $[a,b]$ and let $c \in(a,b)$. Suppose that $\lim_{x\to c}f'(x)=L$ some $L \in\mathbb{R}$. Without using L'Hospital's Rule, prove that $f'(c)=L$. Hint: ...
272 views

### Fixed point in a continuous function

Suppose that $f$ is a function defined in $[a;b]$ to $[a;b]$ and continuous on $[a;b]$. The problem is I haven't the definition of the function, this is more abstract, but even if how can I prove that ...
583 views

### What is $e$? How does $e$ relate to its limit as $n \to \infty$? [closed]

Why does $\left(\frac{\infty + 1}{\infty}\right)^{\infty} = e$? Does this account for the disparity between the countable and uncountable $\infty$? Why?
2k views

### What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...
2k views

111 views

### Finding the limit of roots products $(\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2})$

I need to find: $$\lim_{n \to \infty } (\sqrt{2}-\sqrt[3]{2})(\sqrt{2}-\sqrt[4]{2})(\sqrt{2}-\sqrt[5]{2})\cdot \cdot \cdot (\sqrt{2}-\sqrt[n]{2})$$ So far, I think that ...
691 views

### Solving $\lim\limits_{x\to0} \frac{x - \sin(x)}{x^2}$ without L'Hospital's Rule.

How to solve $\lim\limits_{x\to 0} \frac{x - \sin(x)}{x^2}$ Without L'Hospital's Rule? you can use trigonometric identities and inequalities, but you can't use series or more advanced stuff.
128 views

### Need to show that $\lim_{x\to\infty}\left(\sum_{n\le x}^{}\frac{1}{n}-\ln x \right)$ exist and is less than $1$ [duplicate]

Need some help here. I need prove that the following limit exist and is less than $1$ $$\lim_{x\to\infty}\left(\sum_{n\le x}^{}\frac{1}{n}-\ln x\right)$$ I feel a little lost here, this is my first ...
242 views

### How to prove $\lim_{n \to \infty} (1+1/n)^n = e$?

How to prove the following limit? $$\lim_{n \to \infty} (1+1/n)^n = e$$ I can only observe that the limit should be a very large number! Thanks.
497 views

### What does $\lim\limits_{x \to \infty} f(x) = 1$ say about $\lim\limits_{x \to \infty} f'(x)$?

Given that $f$ is differentiable, what does $\lim\limits_{x \to \infty} f(x) = 1$ say about $\lim\limits_{x \to \infty} f^\prime(x)$ ? Intuitively I feel that it's $0$. I attempted to solve this by ...
253 views

### Finding $\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$ if it exists [duplicate]

Does there exist the following limitation? If the answer is yes, could you show me how to find that? $$\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$$ In the following, I'm going to write what ...
177 views

### Is it trivial to say $\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k}$

Is it trivial to say $$\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k},$$ considering the fact that we know $$\mathop {\lim }\limits_{n \to \infty } {(1 + {1 \over n})^n} = e?$$ ...
696 views

### Sequence of solutions to $x\sin x=1$

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. Consider a sequence $x_n, n\ge1$ formed by positive solutions to ...
952 views

### How to find the sum of the following series

How can I find the sum of the following series? $$\sum_{n=0}^{+\infty}\frac{n^2}{2^n}$$ I know that it converges, and Wolfram Alpha tells me that its sum is 6 . Which technique should I use to ...