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:
$$
\frac{\log_an}{n} &...
1
vote
2answers
55 views
Determine $\lim_{n\to\infty} \left(\left(3\sqrt{n}\right)^{\frac{1}{2n}}\right)$
Determine $\lim_{n\to\infty} \left(\left(3\sqrt{n}\right)^{\frac{1}{2n}}\right)$
Is there a way to determine this limit without using the properties of the logarithm function? Anyways, I am not sure ...
0
votes
2answers
53 views
$\lim_{n \to +\infty} n \cdot\cos^2\left(\frac{n \pi}{3}\right)$
Find $\lim_{n \to +\infty} n \cdot \cos^2\left(\frac{n \pi}{3}\right)$
First I have a look at $\cos(\frac{n \pi}{3})$
What I expect is that the sequence diverges so I want to find two sub-sequences ...
0
votes
4answers
31 views
Find $ lim_{ x \to 0} (\cos(x))^{\cot^2(x)}$ without using L'Hospital [closed]
Is there a way of finding this limit without de L'Hospital (and without series expansion)?
$\lim_{x\to 0} (\cos x)^{\cot^2 x}$
Or this one, which I got trying to solve the first one:
$\lim_{x\to 0} ...
0
votes
1answer
40 views
Evaluate a limit of (e^(-1/x^2))/x [closed]
How to evaluate:
$\lim_{x\to 0} \frac{e^\frac{-1}{x^2}}{x} $
I got the limit from the positive side using squeeze theorem. I don't know how to evaluate limit from the negative side for this.
-5
votes
3answers
57 views
Find $n$ for the given limit.
If
$$\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?
-4
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 ...
