Articles of limits

Determine the limit distribution

I have this question here that I could use some help with. Let $X_1$, $X_2$, . . . be a sequence of random variables such that $P(X_n=\frac{k}{n})=\frac{1}{n}$, for $k=1,2,…,n$ Determine the limit distribution of $X_n$ as $n\rightarrow \infty$. Now I think that if I could find $f_X(x)$ and $F_X(x)$ for $X_n$ then I would have […]

A contradiction or a wrong calculation? $3\lt\lim_{n\to\infty}log_n(p)\lt3$, $\forall n$ (sufficiently large) where $n^3\lt p\in\Bbb P\lt(n+1)^3$?

This is probably a very trivial observation but at the same time somehow interesting to me. For every sufficiently large $n$, it has been already proved that: $$\exists p \in \Bbb P: n^3 \lt p \lt (n+1)^3$$ Then applying $log_n$ two all the terms: $$log_n n^3 \lt log_np \lt log_n(n+1)^3$$ By the properties of logarithms: […]

Can the limit $\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$ be calculated?

$$\displaystyle\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$$ I have this limit to be calculated. Since the first term takes the form $\frac 00$, I apply the L’Hospital rule. But after that all the terms are taking the form $\frac 10$. So, according to me the limit is $ ∞$. But in my book it is given 1/10. How should I solve […]

Limit: $\lim\limits_{x\to{\pi/2}^+} \frac{\ln(x-\pi/2)}{\tan(x)}$ using De l'Hôpital's rule?

Find the limit $$\lim_{x \to {\pi/2}^+} \frac{\ln(x-\pi/2)}{\tan(x)}$$ So, both the numerator and the denominator approach negative infinity. There De l’Hôpital’s rule applies. I found the derivative of both and got: $$\lim_{x \to {\pi/2}^+} \frac{{1/(x-\pi/2)}}{\sec^2(x)}$$ I keep on applying De l’Hôpital’s rule yet still cannot find an expression that will not result in zero as the […]

Limit involving incomplete gamma function

Let $\Gamma(a,x) = \int_x^\infty t^{a-1} e^{-t} dt$ be the incomplete gamma function. What is the limit $$ \lim_{p \to \infty} \frac{1}{\sqrt{p}} \Big[\Gamma(p+\tfrac12,x)\Big]^{\frac{1}{2p}} $$ as a function of $x$?

Does $\lim_{m \to \infty}\sum_{n=1}^m (-1)^n (\sum_{k=n^2}^{(n+1)^2-1}\frac{1}{\sqrt{k}}-2) $ exist?

This question is based on an answer and comment to this question: convergence of $\sum\limits_{n=1}^\infty \frac{(-1)^{\lfloor \sqrt{n}\rfloor}}{\sqrt{n}}$ Does $\displaystyle \lim_{m \to \infty} \sum_{n=1}^m (-1)^n \left[ \sum_{k=n^2}^{(n+1)^2-1}\frac{1}{\sqrt{k}}-2 \right] $ exist? The answers there show that $\sum_{k=n^2}^{(n+1)^2-1}\frac{1}{\sqrt{k}} \to 2 $, but are not precise enough to show that the difference is monotonic, so the alternating series theorem […]

Limit of the composition of two functions with f not necessarily being continuous.

Let $g$ be a continuous function at a point $a \in \mathbb{R}$, and let $$\lim_{x\rightarrow g(a)} f(x) = L$$ Show that $$\lim_{x\rightarrow a} (f\circ g)(x) = \lim_{x\rightarrow g(a)} f(x)$$. How do I prove this in the case that $f$ is not continuous. My current proof is incorrect since it assumes $f$ is continuous. Let $\lim_{u\rightarrow […]

Limit theorem problem

In my calculus book, there are two theorems : (i)$$\lim_{x \rightarrow a}x^n=a^n$$ and (ii) $$\lim_{x\rightarrow a}\frac{x^n-a^n}{x-a}=na^{n-1}$$ $\forall x\in R^+ \forall n\in R.$ In my textbook, the proof of above theorems are given $\forall n\in N$. And told that “We will accept the theorem $\forall n\in R$ and apply.” I am a beginner in calculus.So, I […]

What is the $\lim_\limits{x \to 1+}\left(\frac{3x}{x-1}-\frac{1}{2 \ln(x)}\right)$?

What is the limit of $$\lim_\limits{x \to 1+} \left(\frac{3x}{x-1}-\frac{1}{2 \ln(x)} \right)$$ I attemped the problem using L^Hopital’s Rule. My Work $$\lim_\limits{x \to 1+} \left(\frac{3x}{x-1}-\frac{1}{2 \ln(x)} \right)$$ $$\lim_\limits{x \to 1+} \left(-\frac{3}{(x-1)^2}+\frac{1}{2 \ln^2(x)\cdot x} \right)$$ $$\frac{-3+1}{0}=\frac{-2}{0}$$ The answer is suppose to be $\infty$. I know what I did is probably not right.

Prove that $\{x_1=1,\,x_{n+1}=\frac {x_n}2+\frac 1{x_n} \}$ converges when $n \to \infty$

I want to prove that the sequence defined by $\{x_1=1,\,x_{n+1}=\frac {x_n}2+\frac 1{x_n} \}$ has a limit. By evaluating the sequence I notice that the sequence is strictly monotonically decreasing starting from $x_2=1.5$. It seems to suggest itself to prove that the sequenced is bounded by $1\le\big( x_n \big)_{n\ge1} \le 1.5$ and to prove that it […]