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Questions tagged [limits-without-lhopital]

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

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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}^{\...
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0answers
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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 =...
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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
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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 ...
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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)}...
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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 ...
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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\...
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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 ...
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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}{...
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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^...
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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 ...
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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 ...
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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 ...
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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
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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{...
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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 $$ 2*\lim_{x \to \infty} = n*\sin (\frac{1}{{n}}) $$ ...
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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
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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 ...
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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 ...
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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
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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
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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}{\...
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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/...
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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 ...
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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 ...
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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{...
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2answers
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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 ...
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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 ...
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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 ...
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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: $(a^3-b^3)=(a-b)(a^2+ab+b^2)$ ...
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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 $...
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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^...
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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: $$\...
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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.
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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 ...
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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})...
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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 ...
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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=...
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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
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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
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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 ...