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Questions tagged [limits-without-lhopital]

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

0
votes
3answers
32 views

How do I compute $\lim_{x \to a}(a-x)\tan\left(\frac{πx}{2a}\right)$

Evaluate $$\lim_{x \to a}f(x)=(a-x)\tan\left(\frac{πx}{2a}\right)$$ I tried changing the tangent into cotangent by writing it in the form of $\cot\left(\dfrac{π}{2}-\dfrac{πx}{2a}\right)$. Then I ...
1
vote
1answer
33 views

Limit of sequence $\lim \limits_{n \to \infty\ }(\ln n)^{1+\frac{(\ln(\ln n))^{2019}}{\ln n}}-\frac{\sqrt[2018]n}{\ln n}$

$\lim \limits_{n \to \infty\ }(\ln n)^{1+\frac{(\ln(\ln n))^{2019}}{\ln n}}-\frac{\sqrt[2018]n}{\ln n}$ I can show that $\lim \limits_{n \to \infty\ } {\frac{(\ln(\ln n))^{2019}}{\ln n}}=0 $, so $\...
3
votes
2answers
45 views

Finding a limit involving roots without derivatives

I need to find the following limit $$ \lim_{x\to-1}\frac{1+x^{1/7}}{1+x^{1/5}} $$ using no derivatives. I've tried attempting to rationalize or divide by certain polynomials, but nothing has worked. ...
1
vote
3answers
57 views

When Should I Use Taylor Series for Limits?

I get confused between when to apply L'Hospital Rule and Taylor Series. Is there any set of trigger points in the questions, that would be easier to solve with Taylor Series? For Example, If the ...
-1
votes
4answers
65 views

Does $\lim_{(x,y)\to(0,0)}\frac{6xy^2}{x^2+y^2}$ exist? [on hold]

I am trying to solve the following $$\lim_{(x,y)\to(0,0)}\frac{6xy^2}{x^2+y^2}$$ This is basically $0/0$ form, but as I saw other post that L'Hôpital's rule does not work on multiple variables. ...
1
vote
4answers
78 views

Find limit $\displaystyle \lim\limits _{x\rightarrow \infty }\left(\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} -\sqrt{2x^{4}}\right)$

$\displaystyle \lim\limits _{x\rightarrow \infty }\left(\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}}\displaystyle -\sqrt{2x^{4}}\right)$ $\displaystyle \lim\limits _{x\rightarrow \infty }\left(\sqrt{x^{4} +x^{...
2
votes
1answer
48 views

Chain rule when applying L'Hopital's rule

I have a very basic question regarding derivation function: $$f(\omega(t)) = \frac{2 +x(t)\cdot \frac{d\omega(t)}{dt}}{\omega(t)} $$ when I check for $$= \lim_{\omega(t)\to\ 0}\frac{2 +x(t)\cdot\...
2
votes
2answers
67 views

Proof verification for $\lim_{n\to\infty}\frac{a^n}{(1+a)(1+a^2)\cdots(1+a^n)}$

Find the limit: $$ \lim_{n\to\infty}\frac{a^n}{(1+a)(1+a^2)\cdots(1+a^n)}\ \ \text{for} \ \ a >0 $$ Let: $$ \begin{cases} x_n = \frac{a^n}{(1+a)(1+a^2)\cdots(1+a^n)}\\ n\in\mathbb N \end{cases} ...
2
votes
3answers
48 views

Show that $\lim_{n\to\infty}\frac{\log_an}{n} = 0$ for $0<a<1$

Let $0<a<1$, prove that: $$ \lim_{n\to\infty}\frac{\log_an}{n} = 0 $$ I've started with proving a simpler case for $a>1$. Choose some $\varepsilon >0$ such that: $$ \frac{\log_an}{n} &...
1
vote
2answers
55 views

Determine $\lim_{n\to\infty} \left(\left(3\sqrt{n}\right)^{\frac{1}{2n}}\right)$

Determine $\lim_{n\to\infty} \left(\left(3\sqrt{n}\right)^{\frac{1}{2n}}\right)$ Is there a way to determine this limit without using the properties of the logarithm function? Anyways, I am not sure ...
0
votes
2answers
53 views

$\lim_{n \to +\infty} n \cdot\cos^2\left(\frac{n \pi}{3}\right)$

Find $\lim_{n \to +\infty} n \cdot \cos^2\left(\frac{n \pi}{3}\right)$ First I have a look at $\cos(\frac{n \pi}{3})$ What I expect is that the sequence diverges so I want to find two sub-sequences ...
0
votes
4answers
31 views

Find $ lim_{ x \to 0} (\cos(x))^{\cot^2(x)}$ without using L'Hospital [closed]

Is there a way of finding this limit without de L'Hospital (and without series expansion)? $\lim_{x\to 0} (\cos x)^{\cot^2 x}$ Or this one, which I got trying to solve the first one: $\lim_{x\to 0} ...
0
votes
1answer
40 views

Evaluate a limit of (e^(-1/x^2))/x [closed]

How to evaluate: $\lim_{x\to 0} \frac{e^\frac{-1}{x^2}}{x} $ I got the limit from the positive side using squeeze theorem. I don't know how to evaluate limit from the negative side for this.
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votes
3answers
57 views

Find $n$ for the given limit.

If $$\lim_{x \to 0}\frac{1-\cos3x\cdot\cos9x\cdot\cos27x\cdot…\cdot\cos(3^nx)}{1-\cos\big(\frac{x}{3}\big)\cdot\cos\big(\frac{x}{9}\big)\cdot\cos\big(\frac{x}{27}\big)\cdot…\cdot\cos\big(\frac{x}{3^...
0
votes
3answers
65 views

Proving a second order special limit without derivatives

The special limit $$ \lim_{x \to 0} \frac{e^x-x-1}{x^2}=\frac 1 2 $$ can be proved by Taylor expansion or with L'Hôpital's rule. Is it possoble to prove it without using derivatives?
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votes
1answer
42 views

Calculating $\lim_{x \to \infty} x^{2/3} ( (x+1)^{1/3} - x^{1/3})$ (without l'hopital) [closed]

I need some help calculating this limit: $$\lim_{x \to \infty} x^{2/3} ( (x+1)^{1/3} - x^{1/3})$$ I can't use l'Hopital.
-1
votes
5answers
52 views

Help calculating $\lim_{x \to \infty} \frac{ (1-\frac{15}{x})^{15} -1 }{(1-\frac{15}{x}) -1}$ [closed]

Can someone help me calculating this limit? $$\lim_{x \to \infty} \frac{ (1-\frac{15}{x})^{15} -1 }{(1-\frac{15}{x}) -1}$$ I can't use L'Hospital's rule.
1
vote
2answers
22 views

Limit addition and multiplication question

As x approaches one, the limit of f of x is 6, the limit of g of x is 8, and the limit of h of x is 10. What is the limit as x approaches one of the function f + g times h? So WLOG I just set $f(...
0
votes
4answers
80 views

Evaluate $ \lim_{x\to 0}|\frac{5^x - 5^{-x}}{5^x-1}| $ without using L'Hospital's rule.

$$ \lim_{x\to 0}\left|\frac{5^x - 5^{-x}}{5^x-1}\right| $$ I know the limit is equal to 2. But I am not allowed to use L'Hospital. How can I evaluate the limit without L'Hospital?
0
votes
3answers
55 views

If $\lim \limits_{n \to \infty\ } (a_{n+1}-a_n)=0$ and $(a_{4n})_{n\ge1}$ converges decide whether or not $(a_n)$ converges.

If $\lim \limits_{n \to \infty\ } (a_{n+1}-a_n)=0$ and $(a_{4n})_{n\ge1}$ converges decide whether or not $(a_n)$ converges. My attempt: I think it isn't enough to $a_n$ converges. I know $\lim \...
0
votes
1answer
24 views

Question about separation the limit of a sequence

$a_n=\frac{1000^{\sqrt[100]{n}}}{(1,01)^n}+n^{1000}(0,99)^{\sqrt[100]{n}}$ Is it correct to write that $\lim \limits_{n \to \infty\ }a_n=\lim \limits_{n \to \infty\ } \frac{1000^{\sqrt[100]{n}}}{(1,...
1
vote
2answers
66 views

Finding Limit related to third derivative without L'Hospital

Let $f(x)$ be a real function and we know that $f'(x) , f''(x) , f'''(x)$ are defined on the domain of the function. Find $$\lim_{h\to 0} \frac{f(x+3h) - 3 f(x+h) + 3 f(x-h) - f(x-3h)}{h^3}$$ I can ...
0
votes
1answer
27 views

Trigonometric Limit (No L'Hôpital) [duplicate]

I came across with this limit: $\lim_{x \to 0}\frac{\tan x - \sin x}{x^3}$ I started working it out this way: $\lim_{x \to 0}(\frac{\tan x}{x}\times \frac{1}{x^2}) - (\frac{sin x }{x}\times \frac{1}...
5
votes
5answers
243 views

Limit of $\lim_{x \to 0} (\cot (2x)\cot (\frac{\pi }{2}-x))$ (No L'Hôpital)

$\lim_{x \to 0} (\cot (2x)\cot (\frac{\pi }{2}-x))$ I can't get to the end of this limit. Here is what I worked out: \begin{align*} & \lim_{x \to 0} \frac{\cos 2x}{\sin 2x}\cdot\frac{\cos(\frac{\...
0
votes
8answers
71 views

Limit $\lim_{u\to0} \frac{3u}{\tan 2u}$

I’m currently stuck trying to evaluate this limit, $$ \lim_{u\to0} \frac{3u}{\tan(2u)}, $$ without using L’Hôpital’s rule. I’ve tried both substituting for $\tan(2u)=\dfrac{2\tan u}{1-(\tan u)^2}$, ...
0
votes
2answers
25 views

Finding limit of a strangely defined recursive sequence

I have problems finding limits of recursive sequence like: $a_1 = 5$ $a_{n+1} = \frac{(a_n)^2}{10n + 3} + 1$ I know that intuitively the limit will be $1$, since $\frac{(a_n)^2}{10n + 3}$ tends to 0, ...
1
vote
2answers
40 views

Need a hint evaluating $ \lim\limits_{x\to 0}\frac{x\ln{(\frac{\sin (x)}{x})}}{\sin (x) - x} $

I'm stuck with this. I've tried substituting $t$ for $\frac{\sin (x)}{x}$ and $\sin (x) - x$ but it doesn't work at all. A small hint would be greatly appreciated.
8
votes
5answers
1k views

$\lim\limits_{x\to 0} \frac{\tan x - \sin x}{x^3}$?

$$\lim_{x\to 0} \frac{\tan x - \sin x}{x^3}$$ Solution \begin{align}\lim_{x\to 0} \frac{\tan x - \sin x}{x^3}&=\\&=\lim_{x\to 0} \frac{\tan x}{x^3} - \lim_{x\to 0} \frac{\sin x}{x^3}\\ &=...
0
votes
3answers
22 views

Proof verification of $\lim_{n \to \infty}\frac{q^n}{n} = 0$ for $|q| < 1$ using $\epsilon$ definition

Prove $$ \lim_{n \to \infty}\frac{q^n}{n} = 0 $$ for $|q| < 1$ using $\epsilon$ definition. Using the definition of a limit: $$ \lim_{n\to \infty}\frac{q^n}{n} = 0 \stackrel{\text{def}}{\iff} \...
1
vote
4answers
59 views

What is $\lim_{x \to 3} (3^{x-2}-3)/(x-3)(x+5)$ without l'Hôpital's rule?

I'm trying to solve the limit $\lim_{x \to 3} \frac{3^{x-2}-3}{(x-3)(x+5)}$ but I don't know how to proceed: $\lim_{x \to 3} \frac{1}{x+5}$ $\lim_{x \to 3} \frac{3^{x-2}-3}{x-3}$ = $1\over8$ $\lim_{...
2
votes
4answers
56 views

Calculate the following limit without L'Hopital

$\lim\limits_{x\to 0}\frac{e^x-\sqrt{1+2x+2x^2}}{x+\tan (x)-\sin (2x)}$ I know how to count this limit with the help of l'Hopital rule. But it is very awful, because I need 3 times derivate it. So, ...
0
votes
2answers
31 views

Limit of goniometric function without l'Hospital's rule

I'm trying figure this out without l'Hospital's rule. But I don't know how should I start. Any hint, please? $$\lim_{x\to \frac{\pi}2} \frac {1-\sin x}{\left(\frac\pi2 -x\right)^2 }$$
-1
votes
5answers
53 views

Problems proving that $\lim\limits_{n \to \infty}\frac{2n}{n^3+1}=0 $ [closed]

I have to prove using the definition of a limit. Following the definition I think I should find n for which it holds: $\lvert\frac{2n}{n^3+1}\rvert\lt\epsilon$ But after some transformations I end ...
0
votes
3answers
61 views

Computing limit of $f(x) = x^{(\frac{1}{x} - 1)}$

Let $f(x) = x^{(\frac{1}{x} - 1)}$ . Find $\lim_{x \to 0^{+}} f(x)$ if it exists . My try : $f(x) = x^{(\frac{1}{x} - 1)} = e^{\frac{(1-x) \ln x}{x}}$ . I'm not allowed to use L'Hôpital's rule but ...
1
vote
2answers
71 views

Sandwich Theorem not working?

This is the limit I need to solve: $$\lim_{n \to \infty} \frac{(4 \cos(n) - 3n^2)(2n^5 - n^3 + 1)}{(6n^3 + 5n \sin(n))(n + 2)^4}$$ I simplified it to this: $$\lim_{n \to \infty} \frac{2(4 \cos(n) - ...
1
vote
2answers
50 views

If $\lambda_n = \int_{0}^{1} \frac{dt}{(1+t)^n}$, for $n \in \mathbb{N}$, then $\,\lim_{n \to \infty} (\lambda_{n})^{1/n}=1.$

If $\displaystyle\lambda_n = \int_{0}^{1} \frac{dt}{(1+t)^n}$ for $n \in \mathbb{N}$. Then prove that $\lim_{n \to \infty} (\lambda_{n})^{1/n}=1.$ $$\lambda_n=\int_{0}^{1} \frac{dt}{(1+t)^n}= \frac{2^...
0
votes
2answers
50 views

$\lim \limits_{n \to \infty\ }\sqrt[n^n]{(3n)!+n^n}$

$\lim \limits_{n \to \infty\ }\sqrt[n^n]{(3n)!+n^n}$ $\sqrt[n^n]{(3n)!}\le\sqrt[n^n]{(3n)!+n^n}\le\sqrt[n^n]{(3n)!+(3n)!}=\sqrt[n^n]{2\cdot(3n)!}=\sqrt[n^n]{2}\cdot \sqrt[n^n]{(3n)!}$ But I haven't ...
2
votes
3answers
48 views

Evaluting $\lim_{n \to \infty } \sqrt[n]{25n+n^3}$

$\lim \limits_{n \to \infty\ } \sqrt[n]{25n+n^3}$ $1\leftarrow\sqrt[n]{25n}\le\sqrt[n]{25n+n^3}\le\sqrt[n]{25n^3+n^3}\le \sqrt[n]{26n^3}=\sqrt[n]{26}\cdot\sqrt[n]{n^3}\to1\cdot1=1$ But is it obvious ...
2
votes
4answers
65 views

Evaluate $\lim \limits_{n \to \infty\ } n((n+1)^{\frac{1}{100}}-n^\frac{1}{100})$

$\lim \limits_{n \to \infty\ } n((n+1)^{\frac{1}{100}}-n^\frac{1}{100})$ $n((n+1)^{\frac{1}{100}}-n^\frac{1}{100})=n^\frac{101}{100}((1+\frac{1}{n})^{\frac{1}{100}}-1)=n^\frac{101}{100}(e^{\frac{ln(1+...
-3
votes
1answer
87 views

$\lim \limits_{n \to \infty\ }n(\sqrt[n]n-1)$ [closed]

$\lim \limits_{n \to \infty\ }n(\sqrt[n]n-1)$ How should I start?
3
votes
3answers
75 views

What am I doing wrong finding $\lim_{x\to 0} \left( \frac{1+x\cdot2^x}{1+x\cdot3^x} \right)^{1/x^2}$?

It has an answer here, but I'd like to know where my solution went wrong. $$\lim_{x\to 0} \left( \frac{1+x\cdot2^x}{1+x\cdot3^x} \right)^{\frac{1}{x^2}} $$ $$\lim_{x\to 0} \left( \frac{1+x\cdot2^x +x\...
0
votes
1answer
68 views

Find the limit of $\frac{1}{1-\cos(x)}-\frac{2}{x^2}$ as $x$ approaches $0$

I need to find $$\lim_{x\to 0}\left(\frac{1}{1-\cos(x)}-\frac{2}{x^2}\right)$$ I already found it using Taylor series. However, I'm looking for a solution without Taylor series expansion or L'Hopital'...
-2
votes
3answers
43 views

Evaluate $ \lim \limits_{n \to \infty\ }\frac{1}{n}\sqrt[n]{n^5+(n+1)^5+…+(2n)^5}$

$ \lim \limits_{n \to \infty\ }\frac{1}{n}\sqrt[n]{n^5+(n+1)^5+...+(2n)^5}$ I have no idea. Please give me some hint.
0
votes
5answers
46 views

Proof for sequence $\frac{n^5}{3^n} \rightarrow 0$ [duplicate]

I am improving my skill at formal sequence convergence proofs, I find them very tricky. I want to prove that: $$\frac{n^5}{3^n} \rightarrow0$$ This should be read as "converges to zero", the ...
3
votes
2answers
311 views

How to calculate this limit without L'Hopital rule?

I want to evaluate the following limit without using the L'Hopital rule : $$ \lim\limits_{x\rightarrow 0^+}\frac{e^{x\ln(x)}-1}{x}$$ I know the answer is $-\infty$. I can demonstrate that graphically ...
3
votes
4answers
101 views

Evaluate $\lim \limits_{n \to \infty\ } \Biggl( \frac{2,7}{(1+\frac{1}{n})^n}\Biggr)^n=0?$

$\lim \limits_{n \to \infty\ } \Biggl( \frac{2,7}{(1+\frac{1}{n})^n}\Biggr)^n$ I would like to replace $(1+\frac{1}{n})^n$ by $e$, and then $\frac{2,7}{e}<1$, so $\lim \limits_{n \to \infty\ } \...
2
votes
7answers
83 views

Help calculating $\lim_{x \to \infty} \left( \sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} \right)$

I need some help calculating this limit: $$\lim_{x \to \infty} \left( \sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} \right)$$ I know it's equal to 1 but I have no idea how to get there. Can anyone give ...
0
votes
3answers
35 views

Limit of $\lim_{x \to +\infty} x \left( \sqrt{(1+\frac{a}{x}) (1+\frac{h}{x})} -1 \right)$

Is there a way to calculate this limit? $$\lim_{x \to +\infty} x \left( \sqrt{(1+\frac{a}{x}) (1+\frac{h}{x})} -1 \right)$$ I know it's equal to $\frac{a+h}{2}$, but what method can I use to ...
1
vote
1answer
63 views

Evaluate $\lim \limits_{n \to \infty\ } \sqrt[n]{\left|\frac {1}{n3^n}-\frac {n^{170}}{4^n}\right|}$

$\lim \limits_{n \to \infty\ } \sqrt[n]{\left|\frac {1}{n3^n}-\frac {n^{170}}{4^n} \right|}= \ldots=\lim \limits_{n \to \infty\ } \sqrt[n]{\frac {1}{n3^n}} \cdot\sqrt[n]{\left|1-\left(\frac {3}{4}\...
-3
votes
2answers
76 views

How to calculate $\lim_{x\to 0} \frac{\sin{(\pi \sqrt {x+1} )}}{x}$ without L'Hospital's rule? [closed]

I got stuck trying to calculate the following limits: $$ \lim_{x\to 0-} \frac{\sin{(\pi \sqrt {x+1} )}}{x},\quad \lim_{x\to 0+} \frac{\sin{(\pi \sqrt {x+1} )}}{x} $$ I've tried to approximate the ...