$\{{C_k^n}\}_{k=0}^n$ are binomial coefficients. $G_n$ is their geometrical mean. Prove $$\lim\limits_{n\to\infty}{G_n}^{1/n}=\sqrt{e}$$

Could be the following limit computed without using Stirling’s approximation formula? $$\lim_{n\to\infty} \frac{1}{\sqrt[n+1]{(n+1)!} – \sqrt[n]{(n)!}}$$ I know that the limit is $e$, but I’m looking for some alternative ways that doesn’t require to resort to the use of Stirling’s approximation. I really appreciate any support at this limit. Thanks.

How would I go about proving that $\lim_{n\to\infty}\sqrt{n+1}-\sqrt{n}=0$? I have tried to use Squeeze theorem but have not been able to come up with bounds that converge to zero. Additionally, I don’t think that converting to polar is possible here.

How can you calculate $$\lim_{x\rightarrow \infty}\left(1+\sin\frac{1}{x}\right)^x?$$ In general, what would be the strategy to solving a limit problem with a power?

This question already has an answer here: Limit of $L^p$ norm 2 answers Supremum equal limit of sequence of integrals 1 answer

Prove the following limit without using approximations and derivatives: $$\lim_\limits{x\to -\infty}{\frac{\ln\left(1+3^{x}\right)}{\ln\left(1+2^{x}\right)}}=0$$ I cannot think of any possible factorization or inequality (so that I could use the Squeeze Theorem) that doesn’t use derivatives so as to find this limit. Any hint?

I am working on a homework problem from Avner Friedman’s Advanced Calculus (#1 page 68) which asks Suppose that $f(x)$ is a continuous function on the interval $[0,\infty)$. Prove that if $\lim_{x\to\infty} f(x)$ exists (as a real number), then $f(x)$ is uniformly continuous on this interval. Intuitively, this argument makes sense to me. Since the […]

Consider the following 2-variable function: $$f(x,y) = \frac{x^2y}{x^4+y^2}$$ I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$. I used polar coordinates instead of solving explicitly in $\mathbb R^2 $, and it went as the following: $$ x = r \cos \theta, \qquad y = r\sin\theta $$ Hence, $$\lim_{(x,y) \to (0,0)} […]

This question already has an answer here: Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means 3 answers

Give an example of a sequence $(x_n)$ of real numbers, where $\displaystyle\lim_{n\to+\infty}|x_n-x_{n+1}|=0$, but $(x_n)$ is not a Cauchy sequence

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