Questions tagged [limits-without-lhopital]

The evaluation of limits without the usage of L'Hôpital's rule.

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Answer for the $\lim_{x\rightarrow 0} \frac{x^2}{1-\cos x}$?

So I know that you can multiply by the conjugate and get the correct answer which is 2. However I wanted to know why this method gave me the wrong answer. how I tried to solve it
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2 answers
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Proving $\lim_{x\to0}\sin\left(\frac{\pi}{x}\right)$ is undefined using the $\epsilon$-$\delta$ definition of a limit

It is well known that $$\lim_{x\to0}\sin\left(\frac{\pi}{x}\right)$$ is undefined, which is intuitively true since the function is periodic and oscillates between $1$ and $-1$ as $x$ approaches zero, ...
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Why is $\delta = \epsilon$ for limit of $f(x)=x$ at any point?

[This question is rather a very easy one which I found to be a little bit tough for me to grasp. If there is any other question that has been asked earlier which addresses the same topic then kindly ...
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Solving the limit $\lim_n \left(1+\frac{1}{-n} \right)^{-n}$

I was doing an exercise about limits of sequences and arrived at the following limit: $$\lim_n \left(1+\frac{1}{-n} \right)^{-n}\ \ \ \ (1)$$ We are supposed to solve the limit without using L'hopital'...
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2 votes
2 answers
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Evaluating $\lim_{x\to {0}}\frac{1}{x\arcsin x} - \frac{1}{x^2}$ without L'Hôpital's rule

So I have this limit ... $$\lim_{x\to {0}}\frac{1}{x\arcsin x} - \frac{1}{x^2}$$ Using l'hôpital rule, I know the answer is $-\frac{1}{6}$, but it seems like my professor want me to find another way ...
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Is there a limit which is hard to compute without L'Hôpital's rule?

I know that for limits having $0/0$ or $\infty/\infty$ form, L'Hôpital's rule is a great tool. But usually these problems can be solved without using it like using Taylor series, etc. So I wanted to ...
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1 vote
1 answer
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Can we use $(1+x)^n = 1+nx$ where $x\to0$ and $n$ is $1/0$?

I was solving a limits questions: $$\lim_{x\toπ/4} \tan x^{\tan2x}$$ After putting $x = (π/4)+h$ and solving it I got the expression: $$(1-h/1+h)^{\cot2x} = (1-2h)^{\cot2h}$$ $$(1-2h)^{\cot2h}$$ Now ...
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2 votes
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$L_1=\lim_{x\to 0}\dfrac{1-\cos x\cos 2x \cos 3x}{x^2}\;$ and $L_2=\lim_{x\to 0} \dfrac{1-(\cos x)^{{(\cos 2x)}^{(\cos 3x)}}}{x^2}\;$

If $$L_1=\lim_{x\to 0}\dfrac{1-\cos x\cos 2x \cos 3x}{x^2}\;$$ and $$L_2=\lim_{x\to 0} \dfrac{1-(\cos x)^{{(\cos 2x)}^{(\cos 3x)}}}{x^2},\;$$ then value of $|L_1-L_2|$ is equal to: My Approach: I put ...
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Hint for proving that $\lim_{x \to 0} \log_{10}{|x|}$ does not exist

As the title suggest, i am currently working on an exercise which asks me to prove that $$\lim_{x \to 0} \log_{10}{|x|}$$ does not exist. The proof is via contradiction. My approach so far has been ...
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1 answer
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Evaluating $\lim_{x\to2}\frac{\sqrt[3]{x^2-x-1}-\sqrt{x^2-3x+3}}{x^3-8}$ without Hopital rule

I want to evaluate this limit without applying Hopital rule,$$\lim_{x\to2}\frac{\sqrt[3]{x^2-x-1}-\sqrt{x^2-3x+3}}{x^3-8}$$ After expanding the denominator I got, $$\frac{1}{12}\lim_{x\to2}\frac{\sqrt[...
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7 votes
7 answers
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Elementary proof of $\lim_{n\to\infty}n(\sqrt[n]{n}-1)=\infty$

This question is closely related to this question, but I am not happy with the answers there for several reasons which I will explain in a second. The limit $\lim_{n\to\infty}n(\sqrt[n]{n}-1)=\infty$, ...
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Evaluating $\lim_{x \rightarrow 0} \frac{\sin(\pi x)(1-\cos(\pi x)}{x^2\sin(x)}$ without L'hôpital's rule

I need help finding this limit: $$\lim_{x \rightarrow 0} \frac{\sin(\pi x)(1-\cos(\pi x))}{x^2\sin(x)}$$ I've used L'Hôpital's rule and the solution is $\pi^3/2$. However I'm asked to solved it ...
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5 votes
1 answer
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How do you find this limit with a relationship to $e$ using Taylor series?

The limit in question is $$ \lim_{x \to 0}\left(\frac{\sin(x)}{x}\right)^{1/x^2} $$ When I replace $\sin(x)$ with its Taylor series about $0$ and cancel out the $x$, I get $$ \lim_{x \to 0}\left(1-\...
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1 vote
1 answer
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Limit of sin(sin(sin(x)))

I had an exam with the exercise $$ \lim _{x\to 0}\left(\frac{\sin(\sin(\sin(x)))}{x}\right) $$ but I needed to solve it without using L'hopital rule but I was not sure how to solve it, do you know how ...
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0 votes
1 answer
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Evaluate the common limit [closed]

I'd like more of an explanation than a solution, I'm sorry but I'm studying math again after 12 years, and I don't understand basic concepts. I have to evaluate some limits without using L'Hopital ...
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-2 votes
1 answer
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Solving $\lim_{t \to \infty} t\log(\frac{\frac{\log(\alpha + 1)}{t} - \frac{\log(t + \alpha)}{t}}{ 1 - \frac{1}{t(t + \alpha)}} + 1)$ [closed]

Given the following problem $$\begin{equation} \begin{split} {\label{limit}} \lim_{t \to \infty} t\log\left(\dfrac{\dfrac{\log(\alpha + 1)}{t} - \dfrac{\log(t + \alpha)}{t}}{ 1 - \dfrac{1}{t(t + \...
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1 vote
4 answers
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How to find $\lim_{x\to-\infty} \frac{\sqrt{x^2+2x}}{-x}$?

By factorization: $$\lim_{x\to-\infty} \frac{\sqrt{x^2+2x}}{-x}\tag{1}$$ $$=\lim_{x\to-\infty} \frac{x\sqrt{1+\frac{2}{x}}}{-x}$$ $$=\lim_{x\to-\infty}-\sqrt{1+\frac{2}{x}}$$ If I input $x=-\infty$, ...
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1 vote
2 answers
32 views

Other approaches to evaluate $\lim_{h\to 0} \frac{4^{x+h}+4^{x-h}-4^{x+\frac12}}{h^2}$

$$\lim_{h\to 0} \frac{4^{x+h}+4^{x-h}-4^{x+\frac12}}{h^2}=?$$ I evaluated the limit by using the Hopital rule,$$\lim_{h\to 0} \frac{4^{x+h}+4^{x-h}-4^{x+\frac12}}{h^2}=4^x\lim_{h\to0}\frac{4^h+4^{-h}-...
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3 votes
1 answer
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Limits involving exponents

I don't understand this statement from Wolfram Alpha: Since $5^{2k+1}$ grows asymptotically slower than $3^{4k+1}$ as $k$ approaches $\infty$, $$\lim_{k\to\infty} 3^{-4k-1}\cdot 5^{2k+1} = 0.$$ ...
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  • 465
-1 votes
1 answer
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limit and convergence rate of $\lim_{t\to\infty}\left(1-\frac{\log(ct + c + 1)}{t} - \frac{\log(c+1)}{t}\right)^{t}$

I am trying to compute the limit and the rate of convergence of $$\lim_{t\to\infty}\left(1-\frac{\log(ct + c + 1)}{t} - \frac{\log(c+1)}{t}\right)^{t}$$ where $c$ is a positive constant and $t \in \...
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2 votes
3 answers
107 views

Evaluate $\lim_{n\to\infty} \frac{n!e^n}{n^n}$ with L'Hopital's rule (or without)

The problem is to find the limit $$\lim_{n\to\infty} \frac{n!e^n}{n^n}.$$ My first idea was reorder terms: $$\lim_{n\to\infty}\frac{n!e^n}{n^n}=\lim_{n\to\infty} n!\left(\frac{e}{n}\right)^n$$ with ...
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1 vote
2 answers
77 views

Does this limit $\underset{x\to \pi }{\text{lim}}\frac{\sqrt{1-\cos ^2(x)}}{\sin (x)} $ exist?

I want to compute this limit $$ \underset{x\to \pi }{\text{lim}}\frac{\sqrt{1-\cos ^2(x)}}{\sin (x)} $$ which is one of the indeterminate forms, $\frac00$; Using L'hopital, I get $$ \...
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2 votes
2 answers
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Using the definition of a derivative, solve $\lim_{x \to 0} \frac{(2+h)^{3+h} - 8}{h}$

I can't figure out where to even start, I have looked up the answer on Desmos but it uses L'Hopitals rule which i haven't learned yet. $$\lim_{h \to 0} \frac{(2+h)^{3+h} - 8}{h}$$ I see that i can use ...
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  • 1,063
0 votes
2 answers
61 views

Calculate $\lim_{x \to \infty}{\frac{\log(2x+1)}{\log(3x+2)}}$

Calculate $\lim_{x \to \infty}{\dfrac{\log(2x+1)}{\log(3x+2)}}$ I've used L'Hôpital's rule and the solution is $1$. However, can I calculate it without using L'Hôpital's rule?
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  • 17
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3 answers
64 views

Find $a$ if $\lim_{x\to-2}⁡\frac{3x^2+ax+a+3}{x^2+x-2}$ exists.

Is there a number $a$ such that $$\lim_{x\to-2}⁡\frac{3x^2+ax+a+3}{x^2+x-2}$$ exists? If so, find the value of $a$ I desperately need help in trying to work this out. I already tried out factorising ...
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1 vote
1 answer
91 views

Evaluating $\lim_{x \to +0}\frac{1}{x}{\int^{2022x}_{0}{t\,\sqrt{|\cos(\frac{1}{t})|} \,dt}}$ without L'Hopital's Rule [closed]

My problem is to evaluate the following limit: $$\lim_{x \to +0}\frac{1}{x}{\int^{2022x}_{0}{t\,\sqrt{|\cos(\frac{1}{t})|}\,dt}}$$ I have no idea where to begin.
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0 votes
1 answer
76 views

$\lim_{x\rightarrow +\infty}{8x(x+1)-\sqrt{4x^2+2x}\cdot\sqrt[3]{64x^3+144x^2+90x+17}}$

$$\lim_{x\rightarrow +\infty}{8x(x+1)-\sqrt{4x^2+2x}\cdot\sqrt[3]{64x^3+144x^2+90x+17}}$$ I know that it's $\infty - \infty$, but the square root and cube root make it too complicated. Can anyone help ...
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2 votes
1 answer
135 views

Can I solve this limit in this way and why?

I want to solve this: \begin{equation} L=\lim_{x\rightarrow 0} \frac{\sum\limits_{m=1}^{M}a_{m}\exp\left\{\frac{bm^2}{ (m^2+2x^2)x^2}\right\}{(m^2+2x^2)^{-3/2}}+d_m(m^2+2x^2)^{-1} } {\sum\limits_{m=...
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  • 143
6 votes
2 answers
572 views

Is there a mistake in solving this limit?

I want to solve this: \begin{equation} L=\lim_{x\rightarrow 0} \frac{\sum\limits_{m=1}^{M}a_{m}\exp\left\{\frac{bm^2}{ (m^2+2x^2)x^2}\right\}{(m^2+2x^2)^{-3/2}} } {\sum\limits_{m=1}^{M} c_{m}\exp\...
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  • 143
0 votes
1 answer
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If $\lim_{x\to x_0} \frac{f(x)}{g(x)} =L $ , then what we can say about $\lim_{x\to x_0} \frac{f'(x)}{g'(x)}$? [duplicate]

$f$ and $g$ are differentiable functions so that they have the first derivative in a neighborhood of a point $x_0$, and so that $g(x)\neq0$ and $g'(x)\neq0$ in a neighborhood of a point $x_0$. It also ...
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5 votes
2 answers
169 views

Does the limit $\lim_{x\to 0} \sqrt{x^3 - x^2}$ exist or not?

I am having some arguments with a friend about the following limit: $$\lim_{x\to 0} \sqrt{x^3 - x^2}$$ FACTS: the domain of the function is $x\in \{0\}\cup [1,\ +\infty)$ and $0$ is an isolated point. ...
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1 vote
1 answer
71 views

How to compute $\lim_{x \to 0^+} 1 + \frac{\ln(x)}{x^2}$?

How to compute the following limit? $$\lim_{x \to 0^+} 1 + \frac{\ln(x)}{x^2}$$ I tried factoring $x^2$, $\ln(x)$, tried replacing $x^2$ with $e^{2\ln(x)}$ but nothing could remove the indeterminate ...
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0 votes
1 answer
50 views

How to solve a $\frac{0}{0}$ Limiting Form when L'Hopital's Rule even fails?

Let a function U(T) is given by, $U(T)=\frac{1×e^{-\frac{\epsilon}{k_BT}}+2×e^{-\frac{2\epsilon}{k_BT}}+3×e^{-\frac{3\epsilon}{k_BT}}}{1×e^{-\frac{\epsilon}{k_BT}}+1×e^{-\frac{2\epsilon}{k_BT}}+1×e^{-\...
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-3 votes
1 answer
39 views

$\lim 2^n r^{n^2}$ if $0<r<1$?

I'm trying to find for some $0<r<1$ $$\lim_{n\rightarrow\infty}2^nr^{n^2}.$$ I plugged in a few numbers and know that it should go to zero but I don't know how to justify it using limits.
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  • 1
0 votes
2 answers
53 views

$\lim_{n \to \infty}{a^n \over n!} $. [duplicate]

$$\lim_{n \to \infty}{a^n \over n!}, $$ where $a >1$. I have to solve this using sandwich theorem, and I know this sequence tends to zero. I found the lower bound of this sequence but what about ...
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1 vote
3 answers
73 views

Limit of trigonometrical function $\sin(\pi x)/{(1-x)}$ as $x\to1$?

I have the following function: $$\lim_{x\to1} {\sin(\pi x)\over{1-x}}.$$ I need to calculate the limit, although I can't use here L'Hôpital's rule. I have a clue that says to use a correct ...
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-1 votes
1 answer
125 views

Evaluating $\lim_{x \to \infty}\frac{\sin\frac{1}{x}}{\sin\frac{1}{x}}$ [closed]

Find $\lim_{x \to \infty}\frac{\sin\frac{1}{x}}{\sin\frac{1}{x}}$ According to me, this limit should be one. Am I correct or wrong? And what will be the limit of this function as $x $ tends to zero? ...
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1 vote
1 answer
50 views

Prove a sequence as 1/k as a cluster point and show

I'm going through a set of practice problems on my own and I'm stuck on this one: Let $h(n)$ be the largest prime factor of the integer $n > 1$, and $s(n)$ be the sum of its prime factors, so $h(12)...
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  • 29
-2 votes
1 answer
73 views

find $\lim_{x\to 0} \frac{f(x)}{x^2}$

If $f(x)$ is a function satisfying $f(1+x)+f(1-x)=0$ and $f(x) \geq 0$ for $x \in \Bbb R$ ,then find $\lim_{x\to 0} \frac{f(x)}{x^2}$ I know that the given function is symmetrical about point $x=1$ ...
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0 votes
0 answers
44 views

Limits and Calculus

In my maths textbook, for the theory of limits, there is a statement which I am not able to make intuitive sense of. It says: At a given point, the value of a function and its limit may be different, ...
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-2 votes
2 answers
56 views

How can I solve this without L'Hôpital's rule or Taylor series? [closed]

How can I solve this limit without L'Hôpital's rule or Taylor series?$$\lim_{x\to -1}\frac{\sin(x^3-x)}{x+1}.$$ I was trying to solve this limit but I'm stuck when I multiply it by conjugate of the ...
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1 vote
0 answers
31 views

Limit with l'Hopital

Let $x_1\geq ... \geq x_n = 0$ with $x = (x_1,...,x_n)$, $X =\left\{(x_1,...,x_n) \in [0,1]^n: x_1 \geq ... \geq x_n\right\},$ and $F$ be a twice continuously differentiable function with $F'(0)>0$....
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  • 1,157
0 votes
1 answer
57 views

After apply l'Hopital's rule, when one gets infinity

I am trying to solve this problem : $$ \mathbb{P}=\lim_{x\rightarrow 0} \frac{\frac{p_{H}+(1-p_{H})(1-x)^{t}}{p_{H}+(1-p_{H})(1-x)^{t-x}}-\frac{p_{L}+(1-p_{L})(1-x)^{t}}{p_{L}+(1-p_{L})(1-x)^{t-x}}}{x}...
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1 vote
1 answer
69 views

Find the $\lim_\limits{n\to\infty}{\frac{\sqrt{n^2+1}+\sqrt{n}}{(n^4+1)^{1/4}-\sqrt{n}}}$

How to solve the limits without using L-hospital law, like using rationalisation L-hospital method is taking too long The final answer I got was $$\lim_\limits{n\to\infty}{\frac{\sqrt{n^2+1}+\sqrt{n}}{...
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0 votes
1 answer
41 views

Proof using Epsilon Delta definition of proof. [Read Description]

I need to prove that $$\lim_{(x,y)\to(0,0)} \frac{xy(x^2-y^2)}{x+y} = 0. $$ I checked by approaching origin from all directions by substituting $$ y = mx $$ and doing $$\lim_{(x)\to(0)} \frac{xy(x^2-y^...
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  • 1
0 votes
1 answer
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Limit goes to infinity question: $\lim _{x\to \infty }\left(x^6\left[e^{-\frac{1}{2x^3}}-\cos\left(\frac{1}{x\sqrt{x}}\right)\right]\right)$

$\lim _{x\to \infty }\left(x^6\left[e^{-\frac{1}{2x^3}}-\cos\left(\frac{1}{x\sqrt{x}}\right)\right]\right)$ Any tip on how to calculate it? The solution is :$\frac{1}{12}$ I dont need the way to ...
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0 votes
2 answers
57 views

$\lim f(x)=0, \lim f'(x)=c$, find $\lim \frac{f(x)}{x}$

PROBLEM: Let $f:(0,a)\to \mathbb{R} $ be differentiable. $\lim_{x\to 0+} f(x)=0$ $\lim_{x\to 0+} f'(x)=c$, for some $c\in\mathbb{R}$ Prove that $\lim_{x\to 0+} \frac{f(x)}{x}=c$ SOURCE: Real ...
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5 votes
1 answer
100 views

$x_{n+1}=\log(1+x_n)$, then how can I solve $\lim{nx_n}$ within high-school level?

Given $x_{n+1}=\log(1+x_n)$, I know $x_n\to0$ because if $\lim x_n=\alpha$, then $\alpha=\log(1+\alpha)$. And using Stolz-Cesaro Theorem, then $\lim nx_n=\lim\frac{x_nx_{n+1}}{x_n-x_{n+1}}$, and it ...
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  • 5,099
-1 votes
3 answers
422 views

How to solve $\lim_{x\rightarrow 0} \left( \frac{x+1-\sqrt[2022]{2022x+1}}{x^2}\right)$ without L'hopital's rule? [closed]

$ \displaystyle\lim_{x\rightarrow 0} \left( \frac{x+1-\sqrt[2022]{2022x+1} }{x^2}\right).$
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1 vote
0 answers
20 views

Simple clarification about limits

Ok this is gonna be pretty basic... But I just want to make sure I got this reasoning right. This formula: The last transformation after the multipication/division. This is what's going on, right? (...
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