Questions tagged [limits-without-lhopital]

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

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Solving this limit without L'hopital's

I'm having a problem with limits, I'm just getting started to learn them but we didn't cover the limits of trig functions nor did we cover l'hopital's rule as we didn't cover derivatives yet, how can ...
Kine's user avatar
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Prove/Find limes without L'Hospital [closed]

I would like to understand limits more on the basis of the fundamental definitions. So can someone please help me to prove the following limit without L'Hospital? Unfortunately, I have no approach at ...
RaWa's user avatar
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Limit of the indeterminant form [closed]

I was trying to understand the limit of the following function for $a > 1$ and $0 <b < 1$. $\begin{align} \lim_{x \rightarrow 0} x^b a^{log_2(1/x)} \end{align}.$ This is an indeterminate ...
Celestina's user avatar
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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
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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
55 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
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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'...
Xbz-24's user avatar
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Finding limit as x approches 0 - trigonometry [closed]

which is the limit of (tan(x)/x) as x approaches to 0? without using L'Hopital, how would you get the result
Kuaxina's user avatar
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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
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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
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3 answers
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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
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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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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}{...
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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
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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 ...
Adam Nygren's user avatar
-1 votes
2 answers
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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\...
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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
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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 ...
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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 ...
Adam Nygren's user avatar
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2 answers
56 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
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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
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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
108 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
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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
1 vote
1 answer
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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 ...
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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
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-1 votes
1 answer
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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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Standard limits giving wrong answers.

As we all know, $$ \lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)=1 $$ And I encountered many questions where I used this standard limit and it gave the right answer. However in the following ...
Sanket 's user avatar
1 vote
2 answers
270 views

Solving a limit without using L'Hôpital's rule [closed]

First time posting here. I would like to get some help with this limit. I'm expected to solve it without using L'Hopital's rule, as I haven't been taught said rule, but I'm not sure how to go about it....
Thix's user avatar
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Confusion about Limits (Rationals)

$ f(x)= \begin{cases} x^2&\text{if $x$ is irrational}\,\\ 2x+1&\text{if $x$ is rational}\\ \end{cases} $ I want to calculate the limit of $f(x)$ as $x$ tends to $0$. Is it enough to just say ...
adisnjo's user avatar
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3 answers
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Without using L'Hopitals rule, how do you get $\lim_{x \to 0} \frac{1 - \cos 5x}{\sin 3x}$?

Without using L'Hopitals rule, how do you get the limit $$\lim_{x \to 0} \frac{1 - \cos 5x}{\sin 3x}$$ Using L'Hopital's, I can just take the derivative of the numerator $5 \sin 5x$ and denominator $3 ...
newbie py's user avatar
-1 votes
1 answer
123 views

A limit of $1^\infty$ form (maybe) [duplicate]

$$\lim_{n\to \infty} \left(\frac{(2n)!}{n!n^n}\right)^{\frac{1}{n}}$$ Where I started: For me, this looked like a $1^{\infty}$ form. So I did use the 'monkey-on-the-tree' kind of methodology, where we ...
Harikrishnan M's user avatar
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$\lim_{x \rightarrow 0} \frac{(x^2 - x^3) ((x + 1)^\sqrt{2} - 1)}{\log(1 + 4 x) (1 - \cos(2 x))}$

I'm trying to find the following limit: $$\lim_{x \rightarrow 0} \frac{(x^2 - x^3) ((x + 1)^\sqrt{2} - 1)}{\log(1 + 4 x) (1 - \cos(2 x))}$$ Knowing that: $$ \lim_{x \rightarrow 0} \frac{\log(1 + x)}{x}...
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5 votes
4 answers
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How to solve this limit without using L'Hopital

How to solve this? $$\lim_{h\rightarrow0}\frac{\tan(a+2h)-2\tan(a+h)+\tan a}{h^2}$$ I have solved it by L'Hopital but wasn't able to do it with other method. One method i tried writing $\tan$ as $\sin/...
Macron's user avatar
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18 votes
1 answer
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With $x_1=1$ and $x_{n+1} = \frac{1}{x_1^2+x_2^2+\dots+x_{n}^2}$show$\lim\limits_{n\to\infty}\frac{x_1+x_2+\dots+x_n}{n^{2/3}}=\frac{\sqrt[3]{9}}{2}$

Context I found an extra-hard (but fake) example of a paper which adheres to the syllabus for the hardest mathematics high school course available in the state of NSW in Australia; Mathematics ...
Juan Reight Deag's user avatar
5 votes
1 answer
179 views

If $\lim\limits_{x\to0}f(x)=0$ and $\lim\limits_{x \to 0}\frac{f(2x)-f(x)}{x}=0$ how to rigorously prove that $\lim\limits_{x \to 0}\frac{f(x)}{x}=0$? [duplicate]

I saw this problem on my problem book: IF $\lim\limits_{x\to0 }f(x)=0$ and $\lim\limits_{x \to 0 }\frac{f(2x)- f(x)}{x} =0$ prove that $\lim_{x \to 0} \frac{f(x)}{x}=0 $. I tried to solve it but ...
pie's user avatar
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Nonanalytic function with Taylor series of all orders

I am revising Calculus main concepts, I am with analytic functions. It is a well known fact that a function being analytic implies that admits a Taylor polynomial of arbitrary large order. However, ...
user210089's user avatar
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0 answers
28 views

This limit is puzzling me [duplicate]

I am solving a question which involves finding the continuity of a piece-wise function at a particular point, and it involves finding the below limit:$$\lim_{{x \to 0}} \frac{{\tan x - \sin x}}{{x^3}}$...
Harikrishnan M's user avatar
0 votes
2 answers
105 views

Evaluating $\lim_{x\to 1}\frac{8^x-8}{x-1}$ without using L'Hopital rule

Evaluate without using L'Hopital rule $$\lim_{x\to 1}\frac{8^x-8}{x-1}$$ Help me with this limit.
David David's user avatar
1 vote
0 answers
74 views

Calculating limit of a function without using L'Hospital [closed]

I need to calculate the following limit without using L'Hospital: $$\lim_{x\to\infty} \frac{{\sinh(x) + x + x \ln(x)}}{{e^x}}$$ I am only allowed to use limit rules for sequences (e.g. squeeze theorem)...
thejustin348's user avatar
7 votes
6 answers
212 views

Find $\lim\limits_{x \to 0} \frac{ \sqrt{1+x} - 1} { \sqrt[3]{1+x} - 1}$.

I'm having trouble finding $\lim\limits_{x \to 0} \frac{ \sqrt{1+x} - 1} { \sqrt[3]{1+x} - 1}$. Here's my attempt: $$ \lim_{x \to 0} \frac{\sqrt{1+x} - 1}{\sqrt[3]{1+x} - 1} = \lim_{x \to 0} \frac{\...
Mračna Seka's user avatar
4 votes
2 answers
175 views

Does $\lim_{x \to 1}\left(\frac{x}{[x]}\right)$ exist?

Does the limit $\lim_{x \to 1}\left(\frac{x}{\lfloor x\rfloor}\right)$ exist? Where $\lfloor x\rfloor$ is the Greatest Integer Function or the Floor function. My teacher defined that a limit exists ...
Temp's user avatar
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2 votes
2 answers
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Please identify my mistake: $\lim_{h\to0}\frac{\ln (1+2 h)-2 \ln (1+h)}{h^2}$

Anyone please tell what mistake I am making? $\begin{aligned} \lim_{h \to 0} & \frac{\ln (1+2 h)-2 \ln (1+h)}{h^2} \\ & =\frac{4 \ln (1+2 h)}{4 h^2}-\frac{2 \ln (1+h)}{h^2} \\ & =\frac{4 \...
Dharnesh's user avatar
1 vote
0 answers
22 views

Limit of a function with an integral to have a variable in the upper bound

I am a condensed matter physicist and I am studying a function called Kubo-Toyabe function, the function depends on time $$t$$ and written as follows $$ P_{\mu}^{LF}\left(t\right)=1-\frac{2\Delta^{2}}{...
Jogja_papua's user avatar
3 votes
3 answers
123 views

The value of $\lim_{h\to 0}\frac{\sin(a+3h)-3\sin(a+2h)+3\sin(a+h)-\sin a}{h^3}$ is?

Here was my approach: I grouped the terms $\frac{\sin(a+3h)-\sin a}{h^3} $ and $\frac{3\sin(a+h)-3\sin(a+2h)}{h^3}$. By using first principle and got $\frac{3\cos a-3\cos(a+h)}{h^2}$. However, now ...
Rexquiem's user avatar
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2 votes
5 answers
147 views

Showing $ \lim_{x \to \pi/3} \frac{1-2\cos x}{\sin\left(x-\frac{\pi}{3}\right)}=\sqrt{3} $ without l'Hopital

$$ \lim_{x \to \pi/3} \frac{\left(1-2\cos x\right)}{\sin\left(x-\frac{\pi}{3}\right)} $$ Getting the answer $\sqrt{3}$ is easy with l'Hopital but I want to try to do this in a more formal way. I got ...
Kryptic Coconut's user avatar
0 votes
6 answers
243 views

Evaluation of a limit at infinity [closed]

Evaluate $$\lim_{x \to +\infty} [\sqrt{x}-\ln(x^2+1)]$$ I tried to multiply both numerator and denominator by conjugate and tried applying L'hopital but the calculations become way too complex. Is ...
a_i_r's user avatar
  • 671
5 votes
3 answers
160 views

Solving $\frac{df}{dx} = 2x$ using the definition of a derivative. [closed]

Although it's pretty straightforward to solve the above equation but I was wondering, how about if we use the limit definition of a derivative and then try to solve it. So, the equation looks like $$\...
Tanmay Gupta's user avatar
2 votes
0 answers
72 views

Gamma function limit

I've been trying to solve the following limit: $$ \lim_{{x \to 0^+}} \sqrt{\frac{e^{\Gamma(x)} - e^{-\Gamma(x)}}{\ln(\Gamma(x))}}$$ where $\Gamma(x)$ is the Gamma function and: $$ \lim_{{x \to 0^+}} \...
MicrosoftBruh's user avatar
5 votes
5 answers
154 views

Evaluating $\lim_{x \to 7} \frac{\sqrt{x-3}-2}{\sqrt{x+2}-10+x}$without L'Hopital's rule

Been trying to solve this one on several occasions and can't manage it, my prof disallows the use of L'Hopital's, can someone please explain the trick here? thank you in advance! $$\lim_{x \to 7} \...
thekvant's user avatar

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