Questions tagged [limits-without-lhopital]
The evaluation of limits without the usage of L'Hôpital's rule.
1,865 questions
0
votes
0answers
2 views
Prove that if $\sum_{n=1}^{\infty} z_n = S$ then $\sum_{n=1}^{\infty} \bar{z_n} = \bar{S}$
Prove that if $\sum_{n=1}^{\infty} z_n = S$ then $\sum_{n=1}^{\infty} \bar{z_n} = \bar{S}$
I will make use of the fact that $\sum_{n=1}^{\infty} (x_n + \iota y_n) = X+\iota Y $ $\iff$ $\sum_{n=1}^{\...
0
votes
0answers
10 views
Trouble understanding limit of a sequence of complex numbers, James Brown
I am reading Complex Variables and Applications 8th ed by James Brown and Churchill.
On Pg $183$ Example $2$, it says
Let $z_n=-2+\iota\frac{(-1)^n}{n^2}\;\; n\in N$
Using, $\lim_{n\to \infty} z_n =...
-1
votes
3answers
28 views
Evaluate the limit using power series without L'Hospital's Rule [on hold]
I'm a bit stumped on this one.
Show that $\lim_{x\to0} \frac{e^x -1}{\sin(x)} = 1$ using power series.
The instructions are not to use L'Hospital's Rule. I cannot find a way to do this without L'...
1
vote
1answer
32 views
Solve the following limit as $\lim_{x \to 0}$ [duplicate]
$$\lim_{x \to 0} \frac{x\sin(\sin x) - \sin^2 x}{x^6}$$
**My Attempt: **
I started with L'Hopital's rule. But it quickly became messy. So, I did not continue.
I tried to write the Taylor series of ...
0
votes
1answer
16 views
Complex limits regarding the logarithm
I am relatively new to complex analysis, and in my course we have to calculate the following limits (if they exist):
$\lim \limits_{z \to 1}\frac{log(z)}{z}~~~~~$
$\lim \limits_{z \to -1}\frac{log(z)}...
2
votes
3answers
66 views
Finding $\lim_{x \to \infty} \sqrt{x} c^x$ for $0<c<1$
Is there a short way to prove that $\lim_{x \to \infty} \sqrt{x} c^x = 0$ for $0<c<1$?
I tried using L'Hospital's rule and a few substitutions, and even if I was getting somewhere the proof was ...
-1
votes
2answers
29 views
Help With This Limit! [closed]
I'm trying to solve this limit but i seem to go nowhere or just end up getting odd results. If you were to be a very kind person, could you help me with it. Thanks by the way! Here it is:
$$\lim _{x\...
-3
votes
0answers
41 views
Finding the limit with L'Hôpital's rule (we haven't learnt that yet) where $x$ is not in the equation [closed]
I'm learning pre-calculus, could you help me out here:
$$\lim\limits_{x \to 0} \left(\frac1u - \frac{1}{u^2 + u}\right).$$
I'm suspecting that $u$ is a constant in this case but as it hasn't clearly ...
0
votes
2answers
63 views
Limit as h approaches zero. [closed]
I need some help to do this problem without L'Hospital's Rule. I also get $\frac{0}{0}$
$$\lim_{h\to 0}\frac{\cos(x-2h)-\cos(x+h)}{\sin(x+3h)-\sin(x-h)}$$
And that I have the same problem, $\frac{0}{...
-1
votes
3answers
48 views
Evaluate $\lim_{x\to k}\frac{x^n+ax+b}{x-k}$, where $k$ is a roots of $x^n+ax+b=0$
If $a_1,a_2,..,a_n$ are the roots of the equation $x^n+ax+b=0$ then find $(a_1-a_2)(a_1-a_3)....(a_1-a_n)$
My Attempt
$$
x^n+ax+b=(x-a_1)(x-a_2)(x-a_3)...(x-a_n)=0\\
(x-a_2)(x-a_3)...(x-a_n)=\frac{x^...
3
votes
2answers
29 views
How do I take the limit of this trigonometric function
I was wondering how I could take this limit:
$\lim_{a→1}\frac{\sin(a^4 - 1)}{a^3-1}$
My idea was that if I can get the denominator and the inside of the sin to be the same I can use the sandwich ...
15
votes
6answers
535 views
+50
Very indeterminate form: $\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x \longrightarrow (\infty-\infty)^{\infty}$
Here is problem:
$$\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x$$
The solution I presented in the picture below was made by a Mathematics Teacher
I tried to solve this Limit ...
0
votes
3answers
54 views
Limit without L'Hopital $\lim_{x\to0}\frac{\pi - 4\arctan{1\over 1+x}}{x}$
Evaluate the limit:
$$
\lim_{x\to0}\frac{\pi - 4\arctan{1\over 1+x}}{x}
$$
I've been able to show the limit is equal to $2$ using L'Hopital's rule. After finding the derivative of the nominator ...
0
votes
2answers
60 views
limit of $4x\sin(\frac1x)$
I'm having a bit of a problem taking the limit of functions involving $\sin(\dfrac1x)$.
Mainly, I don't know whether or not L'Hopital's rule is required. Here's the particular problem:
$$\lim\...
1
vote
2answers
34 views
derivative using the definition
find
$$f'(0) $$if$$ f(x)=\sqrt{x+\sqrt{1+x}}$$
so you set up the limit to find the derivative at a given point a
$$\lim\limits_{x \to a } \frac{f(x)-f(a)}{x-a}$$
$$\lim\limits_{x \to 0} \frac{\sqrt{...
0
votes
3answers
38 views
$ 2*\lim_{x \to \infty} = n*\sin (\frac{1}{{n}}) $ without L'Hospital
$$ \lim_{x \to \infty} = 2n*\sin (\frac{1}{{n}}) $$
I have this simple limits and i am confused how to solve it without L'Hopital.
Because we get $$ 2*\lim_{x \to \infty} = n*\sin (\frac{1}{{n}}) $$
...
0
votes
4answers
34 views
trigonometric limit using identities
"find"
$$\lim\limits_{x \to 0} \frac{6x+5x^2}{\tan(4x)}$$
saso what I've tried so far is splitting the $\tan(4x)$ into $\sin(4x)/\cos(4x)$ and try to get to an identity, the ones im allowed to use as ...
5
votes
6answers
101 views
Elementary way to evaluate $\lim_{x\to0}\frac{\sqrt[n]{a+x} - \sqrt[n]{a-x}}{x}$
Evaluate:
$$
\lim_{x\to0}\frac{\sqrt[n]{a+x} - \sqrt[n]{a-x}}{x},\ a>0,\ n\in\Bbb N
$$
I've given it several tries but couldn't find an elementary method to find the limit. Two other ways that ...
0
votes
4answers
89 views
Find $\lim_{x\to0}\frac{\sqrt[5]{2x^2 + 10x + 1} - \sqrt[7]{x^2 + 10x + 1}}{x}$
Evaluate:
$$
\lim_{x\to0}\frac{\sqrt[5]{2x^2 + 10x + 1} - \sqrt[7]{x^2 + 10x + 1}}{x}
$$
I want to find the limit above. So far it seems a good idea to rationalize the nominator in hope for $x$ to ...
0
votes
1answer
64 views
Calculate this limit without L'Hospital's rule or series expansion [duplicate]
Calculating the limit as $x$ approaches $0$ using L'Hospital's rule or series expansion is straightforward, but how to evaluate the limit without either of those techniques.
How to calculate the ...
1
vote
1answer
36 views
Limit Comparison Test with upside-down division
I have a series $\sum_{n=1}^{\infty}a_{n}=\sum_{n=1}^{\infty}\frac{\sqrt[2019]{n+2020}}{n^2-2020}$ and I'm looking for a series that converges in order to use the Limit Comparison Test, such as: $\...
2
votes
2answers
42 views
Solve $\lim_{x \to 0} \frac{\sqrt{1+2x} - \sqrt{1-4x}}{x}$ without L'Hospital's Rule.
I need to solve $\lim_{x \to 0} \frac{\sqrt{1+2x} - \sqrt{1-4x}}{x}$ without using L'Hospital's Rule. Using that rule I found the equation becomes $\lim_{x \to 0}(\frac{1}{\sqrt{1+2x}} - \frac{2}{\...
1
vote
1answer
38 views
Taking a Limit for a Product of Functions Involving Quotient
Consider a linear function $f(x) = ax+1$ which is defined on all $x \neq -1/a$ for some fixed constant $a$ . Given integer $n\geq 1$,
I would like to evaluate the following limit
$$
\lim_{x \to -1/...
3
votes
1answer
64 views
Limits: factoring out $x$ from $\lim_\limits{x\to +\infty}\left(\frac{5-x^3}{8x+2}\right)$
So my teacher said that I cannot use arithmetic operation to factor out $x$ from this type of equation, saying that it's because it's composed only by addition and subtraction. But I don't understand ...
1
vote
4answers
64 views
Evaluate limit without using L'H rule
I was revising teacher's notes about L'H rule and I came across this limit.
$$\lim_{x \to \frac{\pi}{2}} \frac{x-\frac{\pi}{2}}{\sqrt{1-\sin x}}$$
The teacher tried to evaluate without L'H to ...
0
votes
0answers
35 views
Help with Limit 9
I have the following system of partial derivatives:
$$\frac{\partial Y}{\partial K}=\frac{1}{K}\left ( Y-\frac{\partial Y}{\partial L}L \right )$$
$$\frac{\partial Y}{\partial L}=\alpha \left (\frac{...
2
votes
2answers
59 views
Computing $ \lim _{n \rightarrow \infty}\left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^{n}\right]$ [duplicate]
$$
\lim _{n \rightarrow \infty}\left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^{n}\right] \text { equals }\_\_\_\_
$$
I tried to expand the term in power using binomial theorem but still could not ...
-1
votes
5answers
45 views
How can I prove limit of $n^k$ over $c^n$ is 0?
How can I prove that
$$
\lim_{n \to \infty}\frac{n^k}{c^n}=0\ ?
$$
I know it is true by intuition, but I do not know how to prove it.
Here $c\gt1, k\ge1$.
BACKGROUND
I am learning time ...
0
votes
1answer
55 views
Definitionally prove that $\lim_{x \to 0}\frac{f(x)-f(0)}{x^2} = \frac{f''(0)}{2}$
$$\lim_{x \to 0}\frac{f(x)-f(0)}{x^2} = \frac{f''(0)}{2}\quad (f'(0) = 0)$$
It seems quite a rudimentary problem, but I can't find an appropriate solution without using L'hospital's rule and ...
0
votes
7answers
140 views
How to calculate limit in a quotient function
How to calculate the limit of the following function:
$$\lim_{x \rightarrow 1}\frac{\sqrt[3]{2x^2-1} -\sqrt[2]{x} }{x-1}$$
I've tried to use the following formula: $(a^3-b^3)=(a-b)(a^2+ab+b^2)$ ...
0
votes
1answer
41 views
For what real values of $p$ does $\lim_{x\rightarrow\infty}\frac{1}{x^p+x^{-p}}$ exist and is finite?
For what real values of $p$ does exist and is finite the limit $\lim_{x\rightarrow\infty}\frac{1}{x^p+x^{-p}}$?
I know that $\frac{1}{x^p+x^{-p}}\leq\frac{1}{2}$. I think that this happens only for $...
1
vote
2answers
47 views
Show that $e^{cx} \gt | P(x) |$ for some $c \gt 0$
Let $P(x)$ be a polynomial, and let $c > 0$. Show that there
exists a real number $N > 0$ such that $e^{cx} > |P(x)|$ for all $x > N$;
My work so far: $e^{cx}=\sum_{n=0}^{\infty} \frac{cx^...
0
votes
2answers
38 views
Proving a limit equals $0$ using a given theorem
Let $a$ be a real number. Prove that if
$$\lim_{(x, y) \to (0,0)} \frac{ax}{\sqrt{x^2 + y^2}} = 0,$$
then we must have $a = 0$.
I would like to solve this using the following fact:
$$\...
3
votes
2answers
58 views
Limits of exponentials: $\lim_{x \to 1} {x^x - x^{x^x} \over {(1-x)^2}}$
It is a $\frac00$ limit, but I can't seem to figure it out. I tried writing ${x^x}$ as ${e^{x\ln x}} $ and going with L'Hospital from there but I got stuck. Any help is greatly appreciated.
2
votes
1answer
57 views
Prove that $\lim\limits_{(x,y)\to (0,0)} \frac{\sin(xy)}{\sin(x)\sin(y)}$ exists, using $(\epsilon ,\delta)$-argument
I need to prove that $\lim\limits_{(x,y)\to (0,0)} \frac{\sin(xy)}{\sin(x)\sin(y)}$
exists using the $ϵ-δ$ limit definition as $(x,y)→(0,0)$.
I know that the limit exist and is equal to $1$. I worked ...
2
votes
2answers
65 views
Find $\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{n}{k(2n-k+1)}$ and $\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{n}{k(2n-k+1)}-\frac{1}{2}\ln(n)$.
I have to find $\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{n}{k(2n-k+1)}$. This limit is equal to $\lim_{n\rightarrow \infty} \sum_{k=1}^{n}(\frac{\frac{1}{n}}{\frac{k}{n} (2-\frac{k}{n}+\frac{1}{n})...
3
votes
1answer
72 views
Evaluate $\lim\limits_{x \to 0} \dfrac{(ax+1)^b - (bx+1)^a}{x^2}$
Evaluate $$\lim\limits_{x \to 0} \dfrac{(ax+1)^b - (bx+1)^a}{x^2}$$ where $a, b$ are non-negative integers.
I solved this with the help of L'Hopital's rule, but still I'm looking for a different ...
3
votes
7answers
57 views
Another way to compute $\lim\limits_{x\to+\infty}x^2\log\bigg(\dfrac{x^2+1}{x^2+3}\bigg)$
I need to compute as a title a limit with $x\to+\infty$. The only way I found to obtain a result is to use the L'Hôpital's rule:
$$\lim\limits_{x\to+\infty}x^2\log\bigg(\frac{x^2+1}{x^2+3}\bigg)=\lim\...
1
vote
1answer
54 views
Incorrect proof for $\lim_{n \to \infty} c^n = 0$ where $|c|<1$, but not sure why?
Is this short proof valid for $\lim_{n \to \infty} c^n = 0$ where $|c|<1$?
Proof:
Limits are unique, and therefore it is sufficient to find the limit for one value of c. Notice if $c=0$ then $c^n=...
1
vote
4answers
66 views
Evaluate $\lim\limits_{x\to+\infty} \sqrt{x}\left ( \sqrt[3]{x+1}-\sqrt[3]{x-1}\right )$?
How do you evaluate $$\lim\limits_{x\to+\infty} \sqrt{x}\left (\sqrt[3]{x+1}-\sqrt[3]{x-1}\right ) ?$$
Thanks in advance for your support.
2
votes
2answers
75 views
Prove $\lim_{(x, y) \to (0, 0)} f(x)$ exists if and only if $m + n > 2$
Problem:
Let $m, n \in \mathbb{N}$. Show that $\lim_{(x, y) \to (0, 0)} \frac{x^{m}y^{m}}{x^{2} + y^{2}}$ exists if and only if $m + n > 2$.
My try:
I'm really not too sure about how to prove ...
1
vote
3answers
109 views
How can I solve this crazy limit? $\lim _{x\rightarrow0}\frac{1-\cos\left(\frac{1-\cos x \cos 2x}{x^2}-\frac {5}{2}\right)\cos2x}{x^2} $
Firstly, I think this can be done with equivalent infinitesimal, but it seems so much complicated. I'm not very brave to do L'Hospital's rule on this question. And I don't think trig formulas can ...
1
vote
3answers
62 views
Showing that $\lim_{(x,y,z)\to(0,e,-2/3)}\frac{\sqrt[3]{x^4}\left(1-\ln y\right)^\frac 53\left(z+\frac23\right)}{x^4+(y-e)^4+(3z+2)^4}$ does not exist
Show that the following limit does not exist:
Solve using the method described below.
$$\lim_{(x,y,z)\rightarrow (0,e,-2/3)} \frac{\sqrt[3]{x^4}\left(1-\ln(y)\right)^\frac 53 \left(z+\frac23\...
2
votes
1answer
32 views
Applying comparison theorem to limits when inequality is known
I somehow confuse myself whenever I apply the comparison lemma. I know there is the comparison lemma that says the following:
Let the sequence $\{a_{n}\}$ converge to the number $a$. Then the ...
1
vote
3answers
50 views
What real $p$ makes this limit exists and be finite?
The limit is $\lim_{x \to 0} \frac{\sqrt{1+x}-(1+px)}{x^2}$
We must find what real $p$ makes this limit exist and be finite, and determine this limit.
I know it can be done using derivative ideas, ...
2
votes
4answers
84 views
How do I find the limit of $\frac{x \cdot \ln(1+x)}{e^{2x}-1}$
How do I find: $$\lim_{x \to 0} \frac{x \cdot \ln(1+x)}{e^{2x}-1}$$ without using L'Hospital.
I know that $$\lim_{x \to c}\frac{f(x)}{g(x)} = \frac{L}{M}, M \neq 0$$
My problem is I get: $M = 0$
...
1
vote
1answer
36 views
Greatest lower bound of the set : $\{(e^n + 2^n)^\frac1n\ | \; n\in \mathbb{N}\}$
Find the greatest lower bound of the set :
$\{(e^n + 2^n)^\frac1n \mid n\in \mathbb{N}\}$
I will find the limit of $a_n := (e^n + 2^n)^\frac1n $ which if exists, must equal the greatest lower ...
0
votes
0answers
98 views
Convergence of $a_n=\sum\limits_{m=2}^{n} \frac{(-1)^mm}{(\ln(m))^m}$ and $b_n = \sum\limits_{m=2}^{n} \frac{1}{(\ln(m))^m}$
Let $a_1=b_1=0$ and for each $n\geq 2$, let $a_n$ and $b_n$ be real
numbers given by
$a_n=\sum\limits_{m=2}^{n} \frac{(-1)^mm}{(\ln(m))^m}$ and $b_n =
\sum\limits_{m=2}^{n} \frac{1}{(\ln(m))^m}$,...
3
votes
1answer
48 views
$(a_n) $ is a sequence of positive real numbers. The series $\sum a_n$ will converge if
$(a_n) $ is a sequence of positive real numbers. The series $\sum a_n$ will converge if
(a) $\sum a_n^2$ converges.
(b)$\sum \frac{a_n}{2^n}$ converges
(c)$\sum \frac{a_{n+1}}{a_n}$ coverges
(d)$\...
2
votes
1answer
27 views
Let $(a_n),(b_n) $ be sequences of positive real numbers such that $na_n<b_n<n^2a_n$
Let $(a_n),(b_n) $ be sequences of positive real numbers such that $na_n<b_n<n^2a_n$ for all $n\geq2$. If the radius of convergence of $\sum a_n x^n$ is $4$, then $\sum b_n x^n$
A)converges ...
