Articles of limits

Show that $ \lim_{(x,y) \to 0} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \alpha/\gamma + \beta/\delta > 1.$

Ted Shifrin on this site posed an interesting problem to me: show that $$ \lim_{(x,y) \to (0,0)} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \frac\alpha\gamma + \frac\beta\delta > 1, \,\,\,\,\text{where } \alpha, \beta, \gamma, \delta >0$$ I think I’ve got the $(\Leftarrow)$ direction as follows: assume WLOG that $\gamma […]

Find $\,\lim_{x\to 1}\frac{\sqrt{x}-1}{\sqrt{x}-1}$

How do I calculate $\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$. Please help me. Thanks!

The limit of composition of two functions

I need your help in solving this limits problem. Let $f$ and $g$ be two functions defined everywhere. If $\lim_{u\to b} f(u) = c$ and $\lim_{x\to a} g(x) = b$, then you may believe that $\lim_{x\to a} f(g(x)) = c$. This problem shows that this is not always true. Consider functions $f$ and $g$ defined […]

Calculating $\lim\limits_{C \rightarrow \infty} -\frac{1}{C} \log(1 + (e^{-p \gamma C} – 1) \sum\limits_{k=0}^{C}e^{-\gamma C} (\gamma C)^k/k!) $

How to calculate the following limit: $$\lim_{C\rightarrow \infty} -\frac{1}{C} \log\left(1 + (e^{-p \gamma C} – 1) \sum_{k=0}^{C}\frac{e^{-\gamma C} (\gamma C)^k}{k!}\right)$$ Given that $0 \leq \gamma \leq 1$ and $0 \leq p \leq 1$. This is a modified version of a problem I’ve posted earlier here.

Limit involving tetration

Let the notation be $a^{\wedge\wedge}b = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{b\,times}$ for tetration. My mentor conjectured the following: Let $n$ be a positive integer, then let $A(n)$ be any function that satisfies $$\lim\limits_{n\to\infty} \left(e^{\frac1e} + \frac1n\right)^{\wedge\wedge}\left[(10 n)^{1/2} + n^{A(n)} + C+o(1)\right] – n = 0$$ where $C $ is a constant. Then $\lim\limits_{n\to\infty} A(n) = \frac1e $ Conjectured by […]

$\lim_{n\to +\infty}\ e^{\sqrt n } * \left(1 – \frac{1}{\sqrt n}\right)^n$

$$\lim_{n\to +\infty}\ e^{\sqrt n } * \left(1 – \frac{1}{\sqrt n}\right)^n$$ Answers are: A)0 B)1 C)e D)${\sqrt e}$ E)$\frac{1}{\sqrt e}$ I tried working on the second part to get it at a better form. In the end I got $e^{- \sqrt n}$. Returning at the beginning with this new form it would be: $$\lim_{n\to +\infty}\ e^{\sqrt […]

Limit of $\sqrt{x^2-6x+7}-x$ as x approaches negative infinity

What is $\lim\limits_{x\to-\infty}(\sqrt{x^2-6x+7}-x)$ ? Don’t understand how to approach this question

Convert to Riemann Sum

The following limit has to be converted to Riemann Sum. $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{2N}{N^{2}+n^{2}}\right)$$=$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{2}{1+n^{2}/N^{2}}\right).1/N$. Now, replacing $1/N$ by $dx$, $n^{2}/N^{2}$ by $x^{2}$ and summation by integral, we have $$\lim_{N→∞}1/N\sum^{N}_{n=-N}\left(\frac{2}{1+n^{2}/N^{2}}\right)= \lim_{N→∞}[1-(-1)]/N\sum^{N}_{n=-N}\left(\frac{1}{1+n^{2}/N^{2}}\right)$$ $$=2\lim_{N→∞}[1-(-1)]/N\sum^{N}_{n=0}\left(\frac{1}{1+n^{2}/N^{2}}\right)=?$$ I feel that I am very close to the final answer which is $2\int_{-1}^{1} dt/(1+t^{2})$. But I am stuck after this step. please complete […]

Finding supremum of all $\delta > 0$ for the $(\epsilon , \delta)$-definition of $\lim_{x \to 2} x^3 + 3x^2 -x + 1$

In the $(\epsilon , \delta)$-definition of the limit, $$\lim_{x \to c} f(x) = L,$$ let $f(x) = x^3 + 3x^2 -x + 1$ and let $c = 2$. Find the least upper bound on $\delta$ so that $f(x)$ is bounded within $\epsilon$ of $L$ for all sufficiently small $\epsilon > 0$. I know the definition […]

Understanding the solution to $\lim_{n\to\infty}\frac{a^n}{n!}, a>1$

Calculate $$\lim_{n\to\infty}\frac{a^n}{n!}, a>1$$ I need help understanding the solution to this problem: $$a_n:=\frac{a^n}{n!}$$ $$a_{n+1}=\frac{a^{n+1}}{(n+1)!}=\frac{a}{n+1}\cdot \frac{a^n}{n!}=\frac{a}{n+1}\cdot a_n$$ For $$n\geq n_0 :=\left \lfloor{a}\right \rfloor +1>a$$ we have $$\frac{a}{n+1}<\frac{a}{a+1}<1$$ $\Rightarrow$ the sequence $a_n$ is decreasing from $n_0$-th term onwards and obviously $a_n\geq 0, \forall n\in \Bbb N \Rightarrow a_n$ is convergent. Let $L:=\lim_{n\to\infty}a_n$. Then $$a_{n+1}=\frac{a}{n+1}\cdot a_n$$ $$L=\lim_{n\to\infty}\frac{a}{n+1}$$ $$L=0\cdot […]