# 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.

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### Bounding a $\ln(x)/\ln(y)$ expression where L'Hopital's rule doesn't help

How might I upper bound the following within the unit interval as $p\to0$? $$\frac{\ln(p)-\ln\left(\frac{x}{1-x}\right)}{\ln\left(1-\frac{(1-x)\ln(1-x)+x}{x^2}\right)}$$ The quotient is upper ...
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### Given that for each $n,\;x_n^n + x_n-1= 0,$ is $(x_n)_n$ convergent?

Prove that for $n\ge 2$, the equation $x^n + x-1 = 0$ has a unique root in $[0,1]$. If $x_n$ denotes this root, prove that $(x_n)_n$ is convergent and find its limit. The limit is $1$. But to find ...
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### Limit of partial sums where summands also depend on limit index

Let $a_i, a_i^n \in \mathbb{R}_{\geq 0}, i, n \in \mathbb{N}$ such that the series $\sum\limits_{i=1}^\infty a_i$ exists and $\lim\limits_{n \to \infty} a_i^n = a_i$ for all $i$. I would like to prove ...
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### Find the limit of $\mathop {\lim }\limits_{x \to \infty } \left( {{x^p}\left( {\sqrt{{x + 1}} + \sqrt{{x - 1}} - 2\sqrt{x}} \right)} \right)$

Value of p such that $\mathop {\lim }\limits_{x \to \infty } \left( {{x^p}\left( {\sqrt{{x + 1}} + \sqrt{{x - 1}} - 2\sqrt{x}} \right)} \right)$ is some finite | non-zero number. My approach ...
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### Differentiating $\sin x$ using limits

Below i am trying to differentiate $\sin x$ using first principle. $$\lim_{h \to 0} \frac{\sin(x+h)-\sin x}{h}$$ $$=\lim_{h \to 0} \frac{\sin x(\cos h-1)} {h}+\cos x\lim_{h \to 0} \frac{\sin h}{h}$$ ...
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### How to evaluate a limit that contains an integral?

This question is from the JEE Advanced 2007 Paper 1 (Question 49). The question states: $$\lim \limits_{x \to \frac{\pi}{4}} \frac{\int^{\sec^2x}_2 f(t)\ dt}{x^2-\frac{\pi^2}{16}}$$ I tried solving it ...
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### Evaluating $\lim_{(x, y) \to (0, 0)} \frac{xy^2}{x^2-y^2}$ [duplicate]

What is the value of $$\lim_{(x, y) \to (0, 0)} \frac{xy^2}{x^2-y^2}$$ I have already tried the two paths result and I think now that the limit does exist and it is equals to 0. But i really cannot ...
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### Conditional Fatou Lemma [closed]

Exercise If $M_t$ is a local martingale which is almost surely bounded from below by an integrable random variable $X$, show that $M_t$ is a supermartingale. Hint: apply Conditional Fatou Lemma. I ...
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### Evaluate the limit $\lim\limits_{n\rightarrow \infty}\int_{0}^{\pi}\left|\sin(x)-\sin(2nx)\right|\text{ d}x$ [closed]

$$\lim_{n\rightarrow \infty}\int_{0}^{\pi}\left|\sin(x)-\sin(2nx)\right|\mathrm d x$$ Heading I put proper big numbers to $n$ using calculator to estimate the answer. And then I figured out that the ...
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### How to compute $\lim_{x\to{\pi/2}} \frac{\cos{x} }{\sqrt{(1-\sin{x})^2}}$ [closed]

$$\lim_{x\to{\pi/2}} \frac{\cos{x} }{\sqrt{(1-\sin{x})^2}}$$ I am on terms with high school level calculus, and can solve this using L-hospital's method, but cannot come up with any other method to ...
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### Let $f_n \to f$ pointwise. Are there $\varepsilon, N>0$ such that $|f_n (x)| \le |f (x)| + \varepsilon$ for all $n \ge N$ and $x \in X$?

Let $X \neq \emptyset$ and $f, f_n:X \to \mathbb R$ such that $f_n \to f$ pointwise, i.e., $f_n (x) \to f(x)$ for all $x \in X$. I would like to ask if any of below statements is correct. The meaning ...
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### Let $p_n$ be the probability that the figure formed by connecting the four points $O,P,Q,R$ in this order is a convex square.

(Problem) Let rhombus OABC be a rhombus on the plane with one side of length $1$. $\angle A=60^{\circ}$. Let $n$ be a positive integer and $i, j, k$ are integers between $1$ and $n$. Three points P,...
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### prove that $B(\mathbb{R}) = \mathcal{F}(L^1(\mathbb{R}))$

For $p=1,2,$ let $L^p(\mathbb{R})$ be the set of Lebesgue measurable functions $f:\mathbb{R}\to\mathbb{C}$ so that $\lVert f\rVert_p < \infty,$ where $\lVert f\rVert_1 := \int_{\mathbb{R}} |f(x)|dx$...
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### Is the statement $\sum_{j=1}^\infty x_j<\infty,~(x_j\ge0)$ $\Longrightarrow \lim _{k \to \infty} \sum_{j=k}^\infty x_j=0$ true?

As the title states, I would like to know if the statement $$\sum_{j=1}^{\infty} x_{j}<\infty \Longrightarrow \lim _{k \to \infty} \sum_{j=k}^{\infty} x_{j}=0,\qquad x_j\in [0,\infty)$$ is always ...
Calculate $$\lim_{n \to \infty} \bigg(\frac{n}{n^2 + 1} + \frac{n}{n^2 + 2} + \cdots + \frac{n}{n^2 + n}\bigg)$$ Hello! The answer given to this problem is $1$, but I am getting to $0$: Consider $$\... 2 votes 3 answers 26 views ### 2a-1< x+a < 2a+1 \implies |x+a| < 2|a|+1 In tried to do a rigorous delta-epsilon argument for \lim_{x \to a} x^2 = a^2. I found this post, which I uses the same way as I did. But I don't fully understand on of the steps: If |x-a| < 1 ... 0 votes 0 answers 32 views ### How can prove limit of  n\sin(\frac{1}{n})=1  as  n  approaches to infinity? [duplicate] I am trying to solve this sequence limit \lim_{n\to \infty}  n\sin(\frac{1}{n})=1 and I’m struggling finding a bound for |a_n -l| I thought of rewriting |n\sin(\frac{1}{n}) - 1| As |\frac {\... 0 votes 4 answers 73 views ### What is \lim_{x \to 0} \frac{e^{x - \sin x} - 1}{x - \sin x} [closed] Evaluate$$\lim_{x \to 0} \frac{e^{x - \sin x} - 1}{x - \sin x}$$Hello! How do I find the limit without L'Hopital's rule? It is well known that$$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$$So, if the ... 1 vote 1 answer 31 views ### How to prove f(x) = \frac{e^{x^{2}} \sqrt {\sin x}}{\cos x} is continuous on [0,1] using limit theorems? could you please give a hint for this?$$f(x) = \frac{e^{x^{2}} \sqrt {\sin x}}{\cos x}$$I thought to approach breaking the function pieces by taking e^{x^{2}} \sqrt{\sin x} as one function and \... 0 votes 0 answers 20 views ### Definition of sequentially compact subset I am reading "Beginning Functional Analysis" by Karen Saxe, and I have came upon the definition of a sequentially compact subset: Let (M,d) be a metric space. A subset E \subset M is ... 0 votes 3 answers 91 views ### Evaluating \lim _{x\to \infty} \frac{\ln(2x+3)}{\sqrt{x}} [closed] Evaluate this limit$$\lim _{x\to \infty} \frac{\ln(2x+3)}{\sqrt{x}}$$-8 votes 0 answers 37 views ### Solving \lim_{x\to1}\frac{(x+1)\sqrt{x^2-1}}{\sqrt{x^2+3x}-2} without the l'Hopital Rule [closed] This is the limit I'm trying to solve but I'm unable to do so because I always arrive in a 0/0.$$\lim_{x\to1}\frac{(x+1)\sqrt{x^2-1}}{\sqrt{x^2+3x}-2}$$Teacher says the solution is: ∞ Maybe she is ... 7 votes 2 answers 146 views ### show that if \lim (f(x_1)+\cdots +f(x_n))/n exists whenever \lim (x_1+\cdots +x_n)/n exists, then f is continuous Show that if f:[0,1]\to \mathbb{R} and$$\lim \limits _{n\to \infty}\frac{f(x_1)+\cdots + f(x_n)}{n}$$exists whenever$$\lim \limits _{n\to \infty}\frac{x_1+\cdots + x_n}{n}exists, where $(x_n)$ is ...
Suppose that $f: (0,1] → \mathbb{R}$ is such that $\lim \limits_{x \to a^-}f(x) =f(a)$ for all $a∈(0,1]$. Is $f$ continuous on $(0,1]$? I don't really know how to solve this question. I suspect that ...