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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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Evaluate $\lim_{x \to π/4} \frac{\sqrt{2}- \cos(x)- \sin(x)}{(4x-π)^2}$ [closed]

$$\displaystyle \lim_{x \to π/4} \frac{\sqrt{2}- \cos(x)- \sin(x)}{(4x-π)^2}$$ Can anyone pls help me to figure out the solution without using L'Hopital's rule as I thought if there was an alternative ...
LavN's user avatar
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Solving a combined limit with an $1^{\infty}$ form nested inside a 0×∞ form

I came across this limit problem: $\lim _{x \rightarrow \infty}\left\{\left(\frac{x+1}{x-1}\right)^x-e^2\right\} \cdot x^2$ Plugging this into desmos, one can see that the limit approaches $\frac{2 e^...
Afsheen's user avatar
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Do you need L'Hôpital's rule to prove Taylor's formula?

I recently read a Quora answer. The answerer was asked to solve the limit $$\lim_{x\to0}\frac{\cos x-e^x}{\sin x}$$ without using L'Hôpital's rule. The answerer used the Taylor series expansion of the ...
Elvis's user avatar
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2 votes
4 answers
109 views

Is it possible to show that for $q>0$, $\lim\limits_{x\to\infty}\dfrac{(\ln{x)^p}}{x^q} = 0$ without using L'Hopital's Rule?

Is it possible to show that for $q>0$, $\lim\limits_{x\to\infty}\dfrac{(\ln{x)^p}}{x^q} = 0$ without using L'Hopital's Rule? Applying L'Hopital's Rule repeatedly until the numerator becomes a ...
ten_to_tenth's user avatar
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-1 votes
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Limit of square root, not working as expected [duplicate]

Hey i have this function, and I don't understand why I get wrong limit if I insert x into the square root, even though it's correct algebraic to insert it. $$ \frac{\sqrt{x^2 + 9}}{x} $$ The first ...
miiky123's user avatar
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-2 votes
2 answers
135 views

Show that $(1-\frac{1}{n})^{n^2}$ converges to $0$ [duplicate]

I want to show that $\displaystyle\quad\lim_{n \to \infty}\left(1 - {1 \over n}\right)^{n^{2}} = {\large 0}$ This is in a context where L'Hopital isn't allowed, but is known that $$ {\rm e}^{\large a} ...
Benjacort's user avatar
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1 answer
68 views

Restriction on L'Hôpital's rule for oscillating functions as x approaches infinity

Suppose we have, $$\lim_{x\to \infty}\frac{\text{x + sinx}}{\text{x + 2sinx}}$$ When my teacher gave me this problem I could solve it by taking out an $x$ from numerator and denominator: $$\lim_{x\to ∞...
Apoorva Shukla's user avatar
1 vote
1 answer
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Ratio of two diverging integrals

Consider the ratio: $$ r = \frac{\displaystyle\int_{-\infty}^{\infty}dx\, e^{-x^2 / 2a} x^2}{\displaystyle\int_{-\infty}^{\infty}dx\, e^{-x^2 / 2a}} $$ For $a > 0$ we have $r = a$ because after ...
Uri Cohen's user avatar
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Question Regarding I.F. of limits

My calculus class has already gone over indeterminate forms, l'Hospital rule, etc. and I am preparing for an exam. One thing I don't understand in my professor's notes, is a part of the following sum ...
Nate's user avatar
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Continuous differentiability of an exponential function at $x=y=0$

I have a function, $f(x,y) = \sqrt{x^2 + y^2} \exp(-\sqrt{x^2 + y^2}) ~\forall (x,y) \in \mathbb{R}^2$. I have been trying to check whether this function is continuously differentiable (or has bounded ...
Con Vi's user avatar
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2 votes
3 answers
82 views

Finding the limit using sandwich

I got stuck finding : $$\lim_{n\to\infty}\frac{\sqrt[n]{1^n+2^n+ \dots +n^n}}{1+2+\dots+n}$$ here is what i did: $$\underbrace{\frac{1}{n}}_{\underbrace{\to \space0}_{n\to \infty}} = \frac{n}{n^2}=\...
Malka's user avatar
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1 vote
2 answers
76 views

How to calculate the limit $\lim_{x \to 0}\frac{\ln(1 + \sin(12x))}{\ln(1+\sin(6x))}$ without L'Hôpital's rule?

$$ \lim_{x \to 0}\frac{\ln(1+\sin12x)}{\ln(1+\sin6x)} $$ I know it's possible to calculate this limit just by transforming it; I think you need to use the knowledge that $$ \lim_{x \to 0}\frac{\ln(1+x)...
Maciej Miecznik's user avatar
1 vote
1 answer
48 views

sequence of integral of a function

Let $a_n=\frac{1}{n}\int_{0}^{n}\frac{log(2+x)}{\sqrt{1+x}}dx$ then i want to show $\lim_{n\to\infty}a_n\rightarrow 0$. i assume $b_n=\int_{n-1}^{n}\frac{log(2+x)}{\sqrt{1+x}}dx$ and $a_n=\frac{b_1+...
Ricci Ten's user avatar
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1 vote
0 answers
50 views

What even is the notion of a function growing faster than one another? What defines the fastness (or degree) of a function's growth?

The Context: I was reading the book The Road to Reality by Roger Penrose when in his calculus section he gave a paragraph or two about the notion of $C^{\infty}$-smoothness and there was one exercise ...
Kareem Shamma's user avatar
-2 votes
2 answers
70 views

Why am I getting two different answers using different methods?

I was doing this question: $\lim_{x \to 0} \frac{-\log(1+2h)+2\log(1+h)}{h^2}$ Now using L'Hopital's Rule I was easily able to arrive at the answer which is 1. But when I used the formula for limits ...
Madly_Maths's user avatar
2 votes
1 answer
53 views

What's wrong with my solution of James stewart 1.7 Q 36?

Question - Prove that $\lim\limits_{x \to 2} \frac{1}{x} = \frac{1}{2}$ My Solution $|x - 2| < \delta$ $|\frac{1}{x} - \frac{1}{2}| < \epsilon$ $\frac{\delta}{|2(\delta+2)|}<\epsilon\qquad$ ...
androidDeweleper's user avatar
-2 votes
1 answer
86 views

Prove $\lim_{x \to 0}{\frac{f(x)}{x^2}} = \lim_{x \to 0}{\frac{f'(x)}{2x}}$ WITHOUT L'Hospital or Taylor series [closed]

Given that $f(x)$ is differentiable and $f'(x)$ is continuous, $f(0)=f'(0)=0$, $\lim_{x \to 0}{\frac{f'(x)}{x}=1}$, I want to show that $$\lim_{x \to 0}{\frac{f(x)}{x^2}} = \lim_{x \to 0}{\frac{f'(x)}{...
John. P's user avatar
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3 votes
2 answers
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Is this provable? $\lim_{x\to 0} \frac{\sin (\pi \cos^2 x)}{x^2}=\pi $ [duplicate]

I came across this question $$\lim_{x\to 0} \frac{\sin (\pi \cos^2 x)}{x^2}=\pi $$ I tried following method simplify it into a $x\to0$, $\sin x / x$ type limit $$\lim_{x\to 0}\frac{\sin(\pi \cos^2 x)}{...
donthababakka's user avatar
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0 answers
81 views

Conflicting answers when calculating limits [duplicate]

We are asked to evaluate the limit $$\lim_{x \rightarrow \infty}\frac{e^x}{{\left(1+\frac1x\right)}^{x^2}} $$ Applying L'Hospital's rule, we get the correct answer to be $\sqrt e$. However if we apply ...
Eisenstein's user avatar
1 vote
2 answers
130 views

Applying Derivative Formula for $\frac{d}{dx}\frac{1}{\tan x}$

Using the identity, $\cot x = \frac{\cos x}{\sin x}$, I used the derivative formula for limits. This is done as followed: \begin{align} \bigg(\frac{\cos x}{\sin x}\bigg)^{'} &= \lim_{h \to 0} \...
BeaconiteGuy's user avatar
0 votes
2 answers
109 views

Limit of function $\frac{\sin^2(x) - \tan^2(x)}{x^n}$ as $x \to0$

The problem is to find $n$ such that: $$ \lim_{x\to 0} \frac{\sin^2(x) - \tan^2(x)}{x^n} $$ is a non-zero real number. My attempt: The limit is of the type $\left[\frac{0}{0}\right]$. By using L'...
Rupert Rybka's user avatar
1 vote
1 answer
64 views

Noncircular proofs that $\lim_{\theta \to 0} \frac{\sin(\theta)}{\theta}= 1$ [duplicate]

I've been looking over some old calculus stuff, and I came back across the following limit. When deriving $\frac{d}{dx}\sin(x)$, the identity $\lim_{x\to 0}\frac{\sin(x)}{x}=1$ is used as an ...
MattKuehr's user avatar
  • 197
2 votes
3 answers
78 views

proof of limit without L'Hospital's theorem

I am having trouble to prove these two statements, without L'Hospital's theorem. This came from a series convergence test problem, $\sum_{n=1}^\infty \frac{(\ln n)^3}{n^3}$. If this is proven, then I ...
icyrla's user avatar
  • 41
3 votes
1 answer
66 views

Limit with a geometric interpretation

Let $f:ℝ \to ℝ$ be a $C^∞$ curve. Determine the following limit; $$\lim_{x_1 \to x_2} \dfrac{ \int_{x_1}^{x_2} \sqrt{1+f'(x)^2} dx}{\sqrt{(x_2-x_1)^2+(f(x_2)-f(x_1))^2}}$$ My attempt: I recognized ...
Cognoscenti's user avatar
3 votes
2 answers
169 views

Simplification of square roots as denominators with limits

This might be a basic question but it has been the major issue to solving problems I seem to have when it comes to calculus. Consider this limit $$\lim_{\alpha \to \infty}\left[\tanh \alpha\right]= \...
Leonardo's user avatar
0 votes
2 answers
51 views

Limit as $x\to3$ of $\frac{\sqrt{x-2}-1}{\sqrt[6]{x-2}-1}$? [duplicate]

I wanted to found the limit of this function. But I was not capable of obtaining a value for $x\to3 ^-$. However, I did find the value for the other side limit, its equal to $-9$. $f:\mathbb{R} \...
Fcatalan's user avatar
2 votes
3 answers
83 views

Find the limit $\lim_{{x \to 0}} \frac{f(x) e^x-f(0)}{f(x)\cos(x) - f(0)}$

I'm currently working on a set of problems on differentiability, and I'm stuck at the problem I have right here. Suppose that $f$ is differentiable at $x=0$ and that $f'(0) \neq 0$. Find the limit $$ \...
Karl Johan's user avatar
0 votes
0 answers
15 views

Removing Intermediate forms using Integration [duplicate]

As we know that using L'hospital rule rule we can differentiate the numerator and denominator separately to remove the Intermediate form. For example $$\lim_{x\to 0}{\frac{e^x-1}{x^2+x}}$$ So my doubt ...
πααρτθ Σαρθι's user avatar
1 vote
1 answer
39 views

Finding the limit of an equation involving the Gaussian?

Here's a math competition problem that I've been stumped on I tried using L-Hopital's rule, I see how we get the equivalence via the M.V.T., but that approach doesn't seem to be working at all. I'm ...
Flinn Bella's user avatar
4 votes
1 answer
61 views

$\lim\limits_{x\to 0} \dfrac{e^x-e^{x\cos x}}{x+\sin x}$ [duplicate]

$\operatorname{lim}_{x\to 0} \dfrac{e^x-e^{x\cos x}}{x+\sin x}$, L'hopital is not allowed. Divide all limit by $x$ then $\lim\limits_{x\to 0} \dfrac{\dfrac{e^x}{x}-\dfrac{e^{x\cos x}}{x}}{1+\dfrac {\...
Elise9's user avatar
  • 193
0 votes
3 answers
89 views

Evaluating $ \lim_{{x \to 0}} \frac{\cos\left(\frac{\pi}{2 \cos x}\right)}{\sin(\sin x^2)} $ without using L'Hôpital's Rule

I'm working on solving the following trigonometric limit without using L'Hôpital's rule and could use some help: $$ \lim_{{x \to 0}} \frac{\cos\left(\frac{\pi}{2 \cos x}\right)}{\sin(\sin x^2)} $$ I'...
Renato German Chavez Chicoma's user avatar
0 votes
1 answer
48 views

Limit for all $\alpha\in\mathbb{R}$

So I'm in the limit section of a calc 1 book, right before one sided limits, and one of the questions is Find $$\lim_{x\to\infty}x^\alpha(\small\sqrt{\frac{x-1}{x}}-\sqrt\frac{x+1}{x+2})$$$$ \forall\...
Someguyalive's user avatar
3 votes
2 answers
79 views

Compute $r$ such that $\lim_{x\to 0}\frac{3^{\sqrt{x}}-1}{x^r} \neq 0$?

I am trying to compute $r$ such that: $$\lim_{x\to 0}\frac{3^{\sqrt{x}}-1}{x^r} \neq 0$$ I tried a few things, mostly multiplying it by some conjugate and using identities like $e^{\log(x)}=x$ but ...
Red Banana's user avatar
  • 24.2k
0 votes
3 answers
95 views

How to solve $\lim_{x\to0}\frac{\sin x}{\sin(e^{3x}-1)-3x}$ without L'Hopital?

I am not very good at mathematical analysis, I am missing certain gaps and I don't know who to consult. I apologize because I don't know the code to write functions in a more visible way. I have the ...
poison25's user avatar
1 vote
2 answers
185 views

How do we prove that $\lim\limits_{x\to 0} \left(\frac {1+e^{x}}{2}\right) ^{1/x} = \sqrt{e}$?

Wolfram Alpha gives me this solution: $$\lim\limits_{x\to 0} \left(\frac {1+e^{x}}{2}\right) ^{1/x} = \sqrt{e}$$ But I have no idea how to get to that result. I tried using L'Hopital but I found it ...
Atk's user avatar
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0 votes
0 answers
46 views

Limit of $g(x)= x + x^2\exp(\frac{i}{x^2})$ as $x$ approaches $0$

Consider $g(x)= x + x^2 \exp (\frac{i}{x^2}); x \in \mathbb{R}_{+}^{*} $ Now, $x + x^2 \exp (\frac{i}{x^2})=x + x^2 \exp (\frac{1}{-ix^2})$ and: \begin{align*} |\exp(\frac{1}{-ix^2})|=\exp(Re(\frac{1}{...
J P's user avatar
  • 893
1 vote
1 answer
43 views

Limit of function with parameter

Calculate $\lim_{x\rightarrow +\infty}(x+d)^{1+\frac{1}{x}}-x^{1+\frac{1}{x+d}}$ I have no idea what to do here: I tried de l'Hospital, but calculations are just horrible. Any ideas how to even start ...
pueblo30's user avatar
3 votes
1 answer
89 views

Find the limit $\lim_{{x \to \infty}} \left( \sin(\sqrt{x+1}) - \sin(\sqrt{x}) \right)$

I'm working through Advanced Calculus: Theory and Practice by John S. Petrovic and is currently working on problem 3.5.9, which is as follows: Find the limit and give a strict “ε − δ” proof that the ...
Karl Johan's user avatar
-1 votes
2 answers
62 views

Doubt about this limit

I need to calculate $$\lim_{x\to +\infty} \left(e^{1/x} - 1\right)\ln(e^x+1)$$ I thought about using Taylor series in this way $$\lim_{x\to +\infty} \left(1 + \frac{1}{x} + o\left(\frac{1}{x}\right)-1\...
Heidegger's user avatar
  • 3,482
1 vote
1 answer
105 views

Evaluate $\lim\limits_{x \to 0} \frac{\sin(ax) + bx}{\sin(bx)+ax}$ without using L'Hôpital's rule

By using L'Hôpital's rule I can clearly see the answer is $1$. But when I tried without using L'Hôpital's rule, I somehow ended up getting $a/b$ as answer. Here's what I did: $$\begin{align} \lim_{x \...
Hisham Shefeekh's user avatar
-1 votes
3 answers
95 views

Calculate this limit by first principles [closed]

I have to calculate this complex limit by first principles, ie, without using sophisticated tricks like L'Hospital's rule, Stirling formula, gamma function etc. $$ \lim_{n \to \infty} \frac{[(n+1)!]^2 ...
Gustavo's user avatar
  • 2,164
0 votes
2 answers
75 views

The limit of $f(x) = \frac{1}{x^2 + 5x - 24}$ at $x=4$

I'm working through Advanced Calculus: Theory and Practice by John S. Petrovic and is currently working on problem 3.4.2, which is as follows: Find the limit and prove that the result is correct ...
Karl Johan's user avatar
0 votes
2 answers
77 views

Can we use Binomial Approximations when evaluating limits?

I came across this question, Compute $$L=\lim\limits_{x\to 0}{\frac{\sqrt[3]{1+\sin^2 x} \hspace{2mm}-\sqrt[4]{1-2 \tan x}}{\sin x + \tan^2 x}}$$ Can I use Binomial Approximations here? As $x\to 0 \...
Jesko's user avatar
  • 25
2 votes
3 answers
114 views

Proving $\lim_{x\to\infty}xa^{x}=0$ in Elementary Ways

I wish to prove the following limit without using L'Hopital rule or other known limits: $$\lim_{x\to\infty}xa^{x}=0$$ where $0<a<1$. I wanted to do so using this sequence limit (which I know how ...
G.Bar's user avatar
  • 71
1 vote
5 answers
86 views

Is this a correct approach to calculating $\lim_{n\rightarrow \infty} {\sqrt[n]{\ln(n)}}$?

We have just started covering the limit of sequences and I've stumbled upon this limit in our uni's excercises: $$\lim_{n\rightarrow \infty} {\sqrt[n]{\ln(n)}}$$ I've considered solving it using the ...
runtotherescue's user avatar
1 vote
1 answer
120 views

How to compute $\lim_{x\to 0} \frac{e^{ax}-e^{bx}}{x}$?

I'm trying to compute the following limit: $$L=\lim_{x\to 0} \frac{e^{ax}-e^{bx}}{x} \tag{1}$$ And I have to use some of the following limits for it: $$\lim_{x\to 0}(1+x)^{\frac{1}{x}}=e=\lim_{x\to \...
Red Banana's user avatar
  • 24.2k
0 votes
2 answers
108 views

How to compute $\lim_{x\to \infty} x(a^{\frac{1}{x}}-1)$?

I'm supposed to compute it by using some of the following limits somehow: $$\lim_{x\to 0}(1+x)^{\frac{1}{x}}=\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x}$$ But I can't find how to make this limit ...
Red Banana's user avatar
  • 24.2k
1 vote
1 answer
56 views

Limit as $x$ $\to$ $\infty$ of $x\left(1+\frac1x\right)^x-kx^2\ln\left(1+\frac1x\right)$

Evaluate:- $$\lim_{x\to\infty}\left[ x\left(1+\frac1x\right)^x-kx^2\ln\left(1+\frac1x\right)\right]$$ I tried calculating the limits of the two terms separately. By applying L'Hopital, the 2nd limit ...
a_i_r's user avatar
  • 681
0 votes
1 answer
44 views

Can't figure out multivariable limit of $\frac{x^3-x^2y}{x^2+y^6}$ with polar coordinate sub.

I need to find the limit of a function $f(x,y)$ as $(x,y)\rightarrow (0,0)$. The only method I know of is to consider all paths through $(0,0)$ and do polar coordinate substitution to make it into a ...
nanocat's user avatar
  • 35
-1 votes
1 answer
76 views

What is $\lim_{x\to 0} (1+x)e^{-\left(\frac{1}{|x|} + \frac{1}{x}\right)}$ [closed]

$\lim_{x\to 0} (1+x)e^{-\left(\frac{1}{|x|} + \frac{1}{x}\right)}$ Obviously L'hopital is inapplicable here. I guess it can be done by saying that $e^{-\infty}$ is almost zero so the limit is zero ...
Qusai Saify's user avatar

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