Questions tagged [limits-without-lhopital]
The evaluation of limits without the usage of L'Hôpital's rule.
2,803
questions
1
vote
2
answers
49
views
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
0
votes
2
answers
38
views
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, ...
- 21
1
vote
4
answers
92
views
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 ...
- 305
0
votes
1
answer
30
views
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'...
- 3,581
2
votes
2
answers
61
views
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 ...
- 35
1
vote
0
answers
60
views
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 ...
- 196
1
vote
1
answer
62
views
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 ...
2
votes
0
answers
51
views
$L_1=\lim_{x\to 0}\dfrac{1-\cos x\cos 2x \cos 3x}{x^2}\;$ 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 ...
- 2,050
1
vote
0
answers
40
views
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 ...
- 71
3
votes
1
answer
67
views
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,
$$\frac{1}{12}\lim_{x\to2}\frac{\sqrt[...
- 5,676
7
votes
7
answers
415
views
Elementary proof of $\lim_{n\to\infty}n(\sqrt[n]{n}-1)=\infty$
This question is closely related to this question, but I am not happy with the answers there for several reasons which I will explain in a second.
The limit $\lim_{n\to\infty}n(\sqrt[n]{n}-1)=\infty$, ...
- 1,257
1
vote
2
answers
71
views
Evaluating $\lim_{x \rightarrow 0} \frac{\sin(\pi x)(1-\cos(\pi x)}{x^2\sin(x)}$ without L'hôpital's rule
I need help finding this limit:
$$\lim_{x \rightarrow 0} \frac{\sin(\pi x)(1-\cos(\pi x))}{x^2\sin(x)}$$
I've used L'Hôpital's rule and the solution is $\pi^3/2$. However I'm asked to solved it ...
5
votes
1
answer
72
views
How do you find this limit with a relationship to $e$ using Taylor series?
The limit in question is
$$
\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-\...
- 73
1
vote
1
answer
78
views
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 ...
- 179
0
votes
1
answer
24
views
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 ...
- 179
-2
votes
1
answer
65
views
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
$$\begin{equation}
\begin{split} {\label{limit}}
\lim_{t \to \infty} t\log\left(\dfrac{\dfrac{\log(\alpha + 1)}{t} - \dfrac{\log(t + \alpha)}{t}}{ 1 - \dfrac{1}{t(t + \...
- 35
1
vote
4
answers
82
views
How to find $\lim_{x\to-\infty} \frac{\sqrt{x^2+2x}}{-x}$?
By factorization:
$$\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}}$$
If I input $x=-\infty$, ...
- 2,457
1
vote
2
answers
32
views
Other approaches to evaluate $\lim_{h\to 0} \frac{4^{x+h}+4^{x-h}-4^{x+\frac12}}{h^2}$
$$\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}-...
- 5,676
3
votes
1
answer
38
views
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 $\infty$,
$$\lim_{k\to\infty} 3^{-4k-1}\cdot 5^{2k+1} = 0.$$
...
- 465
-1
votes
1
answer
51
views
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 \...
- 35
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 ...
- 23
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
$$ \...
- 165
2
votes
2
answers
57
views
Using the definition of a derivative, solve $\lim_{x \to 0} \frac{(2+h)^{3+h} - 8}{h}$
I can't figure out where to even start, I have looked up the answer on Desmos but it uses L'Hopitals rule which i haven't learned yet.
$$\lim_{h \to 0} \frac{(2+h)^{3+h} - 8}{h}$$
I see that i can use ...
- 1,063
0
votes
2
answers
61
views
Calculate $\lim_{x \to \infty}{\frac{\log(2x+1)}{\log(3x+2)}}$
Calculate $\lim_{x \to \infty}{\dfrac{\log(2x+1)}{\log(3x+2)}}$
I've used L'Hôpital's rule and the solution is $1$. However, can I calculate it without using L'Hôpital's rule?
- 17
0
votes
3
answers
64
views
Find $a$ if $\lim_{x\to-2}\frac{3x^2+ax+a+3}{x^2+x-2}$ exists.
Is there a number $a$ such that
$$\lim_{x\to-2}\frac{3x^2+ax+a+3}{x^2+x-2}$$
exists? If so, find the value of $a$
I desperately need help in trying to work this out. I already tried out factorising ...
- 41
1
vote
1
answer
91
views
Evaluating $\lim_{x \to +0}\frac{1}{x}{\int^{2022x}_{0}{t\,\sqrt{|\cos(\frac{1}{t})|} \,dt}}$ without L'Hopital's Rule [closed]
My problem is to evaluate the following limit:
$$\lim_{x \to +0}\frac{1}{x}{\int^{2022x}_{0}{t\,\sqrt{|\cos(\frac{1}{t})|}\,dt}}$$
I have no idea where to begin.
- 29
0
votes
1
answer
76
views
$\lim_{x\rightarrow +\infty}{8x(x+1)-\sqrt{4x^2+2x}\cdot\sqrt[3]{64x^3+144x^2+90x+17}}$
$$\lim_{x\rightarrow +\infty}{8x(x+1)-\sqrt{4x^2+2x}\cdot\sqrt[3]{64x^3+144x^2+90x+17}}$$
I know that it's $\infty - \infty$, but the square root and cube root make it too complicated. Can anyone help ...
- 341
2
votes
1
answer
135
views
Can I solve this limit in this way and why?
I want to solve this:
\begin{equation}
L=\lim_{x\rightarrow 0} \frac{\sum\limits_{m=1}^{M}a_{m}\exp\left\{\frac{bm^2}{ (m^2+2x^2)x^2}\right\}{(m^2+2x^2)^{-3/2}}+d_m(m^2+2x^2)^{-1} } {\sum\limits_{m=...
- 143
6
votes
2
answers
572
views
Is there a mistake in solving this limit?
I want to solve this:
\begin{equation}
L=\lim_{x\rightarrow 0} \frac{\sum\limits_{m=1}^{M}a_{m}\exp\left\{\frac{bm^2}{ (m^2+2x^2)x^2}\right\}{(m^2+2x^2)^{-3/2}} } {\sum\limits_{m=1}^{M} c_{m}\exp\...
- 143
0
votes
1
answer
62
views
If $\lim_{x\to x_0} \frac{f(x)}{g(x)} =L $ , then what we can say about $\lim_{x\to x_0} \frac{f'(x)}{g'(x)}$? [duplicate]
$f$ and $g$ are differentiable functions so that they have the first derivative in a neighborhood of a point $x_0$, and so that $g(x)\neq0$ and $g'(x)\neq0$ in a neighborhood of a point $x_0$.
It also ...
- 149
5
votes
2
answers
169
views
Does the limit $\lim_{x\to 0} \sqrt{x^3 - x^2}$ exist or not?
I am having some arguments with a friend about the following limit:
$$\lim_{x\to 0} \sqrt{x^3 - x^2}$$
FACTS: the domain of the function is $x\in \{0\}\cup [1,\ +\infty)$ and $0$ is an isolated point.
...
1
vote
1
answer
71
views
How to compute $\lim_{x \to 0^+} 1 + \frac{\ln(x)}{x^2}$?
How to compute the following limit?
$$\lim_{x \to 0^+} 1 + \frac{\ln(x)}{x^2}$$
I tried factoring $x^2$, $\ln(x)$, tried replacing $x^2$ with $e^{2\ln(x)}$ but nothing could remove the indeterminate ...
- 21
0
votes
1
answer
50
views
How to solve a $\frac{0}{0}$ Limiting Form when L'Hopital's Rule even fails?
Let a function U(T) is given by,
$U(T)=\frac{1×e^{-\frac{\epsilon}{k_BT}}+2×e^{-\frac{2\epsilon}{k_BT}}+3×e^{-\frac{3\epsilon}{k_BT}}}{1×e^{-\frac{\epsilon}{k_BT}}+1×e^{-\frac{2\epsilon}{k_BT}}+1×e^{-\...
- 23
-3
votes
1
answer
39
views
$\lim 2^n r^{n^2}$ if $0<r<1$?
I'm trying to find for some $0<r<1$
$$\lim_{n\rightarrow\infty}2^nr^{n^2}.$$
I plugged in a few numbers and know that it should go to zero but I don't know how to justify it using limits.
- 1
0
votes
2
answers
53
views
$\lim_{n \to \infty}{a^n \over n!} $. [duplicate]
$$\lim_{n \to \infty}{a^n \over n!}, $$ where $a >1$.
I have to solve this using sandwich theorem, and I know this sequence tends to zero.
I found the lower bound of this sequence but what about ...
- 165
1
vote
3
answers
73
views
Limit of trigonometrical function $\sin(\pi x)/{(1-x)}$ as $x\to1$?
I have the following function:
$$\lim_{x\to1} {\sin(\pi x)\over{1-x}}.$$
I need to calculate the limit, although I can't use here L'Hôpital's rule.
I have a clue that says to use a correct ...
- 51
-1
votes
1
answer
125
views
Evaluating $\lim_{x \to \infty}\frac{\sin\frac{1}{x}}{\sin\frac{1}{x}}$ [closed]
Find $\lim_{x \to \infty}\frac{\sin\frac{1}{x}}{\sin\frac{1}{x}}$
According to me, this limit should be one. Am I correct or wrong?
And what will be the limit of this function as $x $ tends to zero? ...
- 5,502
1
vote
1
answer
50
views
Prove a sequence as 1/k as a cluster point and show
I'm going through a set of practice problems on my own and I'm stuck on this one:
Let $h(n)$ be the largest prime factor of the integer $n > 1$, and $s(n)$ be the sum of its prime factors, so $h(12)...
- 29
-2
votes
1
answer
73
views
find $\lim_{x\to 0} \frac{f(x)}{x^2}$
If $f(x)$ is a function satisfying $f(1+x)+f(1-x)=0$ and $f(x) \geq 0$ for $x \in \Bbb R$ ,then find $\lim_{x\to 0} \frac{f(x)}{x^2}$
I know that the given function is symmetrical about point $x=1$ ...
- 5,502
0
votes
0
answers
44
views
Limits and Calculus
In my maths textbook, for the theory of limits, there is a statement which I am not able to make intuitive sense of. It says:
At a given point, the value of a function and its limit may be different, ...
-2
votes
2
answers
56
views
How can I solve this without L'Hôpital's rule or Taylor series? [closed]
How can I solve this limit without L'Hôpital's rule or Taylor series?$$\lim_{x\to -1}\frac{\sin(x^3-x)}{x+1}.$$
I was trying to solve this limit but I'm stuck when I multiply it by conjugate of the ...
1
vote
0
answers
31
views
Limit with l'Hopital
Let $x_1\geq ... \geq x_n = 0$ with $x = (x_1,...,x_n)$, $X =\left\{(x_1,...,x_n) \in [0,1]^n: x_1 \geq ... \geq x_n\right\},$ and $F$ be a twice continuously differentiable function with $F'(0)>0$....
- 1,157
0
votes
1
answer
57
views
After apply l'Hopital's rule, when one gets infinity
I am trying to solve this problem :
$$ \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}...
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
$$\lim_\limits{n\to\infty}{\frac{\sqrt{n^2+1}+\sqrt{n}}{...
0
votes
1
answer
41
views
Proof using Epsilon Delta definition of proof. [Read Description]
I need to prove that
$$\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^...
- 1
0
votes
1
answer
49
views
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 ...
user1028660
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 ...
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 ...
- 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).$
- 19
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? (...
- 221