Questions tagged [limits-without-lhopital]

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

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A Question on limits from differential calculus for beginners

$\lim_{x\rightarrow 0} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^3}}$ Could anyone please tell me how to solve this using the Maclaurin series?
Chatresh Ramasai Gudi's user avatar
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17 views

Did I apply the squeeze law in this multivariate limit question correctly?

I am familiar with the squeeze law being used for a single-variable limit context, but was wondering if it can be applied to the following multivariable limit too: $$\lim_{(x,y)\to(0,0)}\frac{x^2y^3\...
Holland Davis's user avatar
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2 answers
107 views

This is a problem from Differential Calculus for Beginners by Joseph Edwards [duplicate]

$$lim_{x\rightarrow 0} \left(\frac{\sin x}{x}\right)^{\frac{1}{x}}$$ I attempted to expand sin x and am unsure how to proceed. $$1-\frac{x}{3!}\left[x^2-\frac{x^4}{5\times 4} \cdots \right]$$ Can ...
Chatresh Ramasai Gudi's user avatar
2 votes
3 answers
85 views

If $|f'(x)-e^{2x}|\leq 3$ for all $x\in\mathbb{R}$. Then evaluate the limit $\lim_{x\to +\infty}\frac{f(x)}{e^{2x}}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that $f'$ is continuous and satisfies $|f'(x)-e^{2x}|\leq 3$ for all $x\in\mathbb{R}$. Then evaluate the limit $$\lim_{x\to +\infty}\...
Maverick's user avatar
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Prove that the limit does not exist: $\lim_{(x,y)\to(1,3)}\frac{|\sin(3x-y)|}{\sqrt{2(x-1)^2+5(y-3)^4}}$ (Without L'Hopital's)

It seems that the function is underfined at $(1,3)$, so when I try to directly plug in $(1,3)$ via limit laws, I get an undetermined term $\frac{0}{0}$. So I tried instead to consider individual paths ...
Jason Xu's user avatar
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Limit of an exponential function can not use L'Hopital rule [closed]

I am stuck on a question about limits, and I must not use L'Hôpital rule. $$\lim_{x\to0}=\frac{9^x-5^x}{4^x-3^x}$$ I don't have any idea to solve this. Can anyone help me please? I appreciate it a lot!...
codemaivanngu's user avatar
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48 views

Taylor series and integral [closed]

If we know Taylor's series of $csch^{2}(x)$ as below (please look at https://www.wolframalpha.com) $$csch^2x= {\frac{1}{x^2}-\frac{1}{3}+\frac{x^2}{15}-\frac{2x^4}{189}+\frac{x^6}{675}-...}$$ Can ...
Riva's user avatar
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2 votes
2 answers
125 views

Evaluating a Limit (without L'Hopital's rule) [duplicate]

I just got this limit problem. It is easy but there is an instruction to solve this limit without L'Hopital's rule. How can I derived this? $$\lim_{x\to 0} \left(\frac{1}{\sin^2(x)} + \frac{\cos^2(x)}{...
leouisscioust's user avatar
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1 answer
137 views

Evaluating $\lim_{x\to\infty}\frac{\ln\cos\frac {48} x}{\ln\cos\frac{1}{12x}}$ without using L'Hopital

I've seen some similar question, but none with $x\to\infty$ $$\lim_{x\to\infty}\frac{\ln\cos\frac {48} x}{\ln\cos\frac{1}{12x}}$$ one of the ideas that i tried is, with $t=\frac{48}{x}$, changing the ...
Green's user avatar
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3 votes
3 answers
103 views

Evaluating $\lim_{x\to0}\frac{\sqrt{\frac{1}{\cos x}}-1}{\sin^2 {\frac{x}{16}}}$ without L'Hopital's Rule

I was solving the limit $$\lim_{x\to0}\frac{\sqrt{\frac{1}{\cos x}}-1}{\sin^2 {\frac{x}{16}}}$$ I simplified the numerator by multiplying both the numerator and denominator by $\sqrt{\frac{1}{\cos x}}+...
Green's user avatar
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3 answers
56 views

Why doesn't L'Hôpital's Rule work on $\lim_{T\to\infty} \frac{\frac12T-\frac{1}{4}\sin2T}{T}$? [duplicate]

$$\lim_{T\to\infty} \frac{\frac{T}{2}-\frac{1}{4}\sin2T}{T}$$ If I solve this limit by breaking it into 2 parts and consider $\lim_{T\to\infty} \frac{sin2T}{T}=0$ and $\lim_{T\to\infty}\frac{T/2}{T}=\...
Aditya Mukherjee's user avatar
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1 answer
53 views

What is infinite limit of rational trigonometric function? [closed]

How can we justify the following limit? $$ \lim _{x \rightarrow -\infty} \dfrac{1+\cos x}{3+2\cos x} $$ We can see the result of limit DNE or $[0, 2]$ on the graph. How can we show?
Ramazan Özarslan's user avatar
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show that $\lim_{x\rightarrow-\infty}\frac{1}{x^k}=0$

I am reading Calculus, Purcell and there is a solved example Show that if $k$ a positive integer, then $$\lim_{x\rightarrow\infty}\frac{1}{x^k}=0\ \text{ and} \ \lim_{x\rightarrow-\infty}\frac{1}{x^k}=...
SMOK Z's user avatar
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2 votes
1 answer
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Applying the limit definition of a derivative on a radical function $x^{2/3}$

I'm trying to find the derivative of the following using the limit definition of a derivative: $$f(x)=x^{2/3}.$$ I know that the derivative of $f(x)$ is $\frac23x^{-1/3}$ by the power rule, but I can'...
Hannah Kelley's user avatar
2 votes
0 answers
47 views

Evaluate $\lim_{x\to \infty} \left(1+\frac{1}{x^2} \right)^{x}$ without L'Hôpital. [duplicate]

Evaluate $\displaystyle\lim_{x\to \infty} \left(1+\dfrac{1}{x^2} \right)^{x}$ without l'hôpital. What I have tried: $\displaystyle\lim_{x\to \infty} \left(1+\dfrac{1}{x^2} \right)^{x} = \displaystyle\...
GaussCauchy's user avatar
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0 answers
44 views

Find $\lim_{{n \to \infty}} \frac{{2^n + a^n}}{{3^n + b^n}}$, knowing that a and b are positive real numbers. [duplicate]

Evaluate $$\lim_{{n \to \infty}} \frac{{2^n + a^n}}{{3^n + b^n}}$$, $a, b \in \mathbb{R}^+$. Found this limit in an old high-school book. Tried every method provided in the chapter, unsuccessfully. ...
rieshy's user avatar
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1 answer
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How do I evaluate whether a limit exists or not without a graph?

I am having a hard time evaluating when a limit exists. I know that for the two-sided limit $\lim_{x\to a} f(x)$ to exist,both $\lim_{x\to a^+} f(x)$ and $\lim_{x\to a^-} f(x)$ must be equal, but it ...
Paul's user avatar
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3 votes
2 answers
149 views

Limit of the sequence $(2a^{\frac{1}{n}}-1)^n$, $a>1$. Without using L'hôpital rule or continuity.

Using L'hôpital rule and continuity of exponential function, we can show that For any $a>1$ $$\lim_{x \to \infty} (2a^{\frac{1}{x}}-1)^x = a^2$$ Since the sequence is just restriction on domain to $...
niraj panakhaniya's user avatar
-1 votes
2 answers
79 views

What is the importance of small angle approximation [closed]

I have seen many approximations like $$\tan x \approx x$$ $$\sin x \approx x$$ $$\cos x \approx 1-\dfrac{x^2}{2}$$ $$\dfrac{\cos^2(x)}{\sin(x)\tan(x)} \approx \dfrac{x^2}{4} + x^{-2} - 1 $$ These are ...
Adhithyan A S's user avatar
1 vote
6 answers
156 views

Evaluating $\lim_{x\to-\infty}\ln(1+\frac{1}{x})-\frac{1}{x+1}$ with sign information

Let $g(x)$= $\ln(1+\frac{1}{x})-\frac{1}{x+1}$ Clearly, $\lim\limits_{x\to+\infty} g(x) = 0^{+}$ I am having trouble wrapping my head around $\lim\limits_{x\to-\infty} g(x) $ We know for $|\frac{1}{x}|...
mathenjoyer's user avatar
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1 answer
91 views

How do we prove that the slope of $f(x) = \cos(x)$ is $0$ at $x = 0$ without using the fact that $\frac{d}{dx} \cos(x) = -\sin(x)$?

In the proof of $\frac{d}{dx} \sin(x) = \cos(x)$ using the first principle of derivatives: $\lim_{\Delta{x} \to 0}\frac{\sin(x+\Delta{x})-\sin(x)}{\Delta{x}} = \lim_{\Delta{x} \to 0}\frac{\sin(x)\cos(\...
All is number's user avatar
-1 votes
2 answers
47 views

Sign of the limit of $x^β (-\log x)^α$ as $x \to 0^+$

My analysis book [1] explains that $$ \lim_{x \to +\infty} \frac{(\log_a x)^α}{x^β} = 0 $$ for $β > 0$, and that by replacing $x = 1/y$ we get $$ \lim_{y \to 0^+}y^β(-\log_a y)^α = 0.\tag{*}\label{*...
Arch Stanton's user avatar
1 vote
2 answers
107 views

Two possible answers for α and β

So there was this limits question in the JEE Main 2022 exam: If $\lim\limits_{n \to \infty}( \sqrt{n^{2}-n-1}+n\alpha +\beta)=0$ Find $8(\alpha + \beta)$ It had the following options: A. 4 B. -8 C. 8 ...
mathenjoyer's user avatar
-1 votes
1 answer
48 views

How to solve $\lim\limits_{x\to 1}\frac{\sin \pi x}{x-1}$ without L'Hospital [closed]

$\lim\limits_{x\to 1}\frac{\sin \pi x}{x-1}$ Been struggling solving this without L'Hospital
Matt Anantachina's user avatar
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0 answers
49 views

Limit of $x[3−\cos(x^2)]$ using the epsilon-delta definition

This is a duplicate of this post here, but I'm new to Stack Exchange so I'm not able to comment on that post yet (and sorry in advance if this question isn't appropriate to the platform). I was ...
cb2's user avatar
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1 vote
2 answers
78 views

trigonometric limit factoring

I have this problem that I can't solve: $$\lim_{x \to 0^+} \frac{3x-\sin 2x}{\sqrt{1-\cos x}}$$ Tried to multiply by the conjugate, so I can get rid of the squareroot on the denominator, but I am ...
TomiMP's user avatar
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3 votes
3 answers
95 views

Evaluate $ \lim_{n \to \infty} \sin^n\left(\frac{2\pi n}{3n+1}\right)$

I'm trying to solve this limit rigorously $ \lim_{n \to \infty} \sin^n\left(\frac{2\pi n}{3n+1}\right)$ I can see that the answer is $0$ as $$\lim_{n \to \infty} \sin\left(\frac{2\pi n}{3n+1}\right) ...
grey's user avatar
  • 400
7 votes
3 answers
234 views

Evaluate $\lim\limits_{x\to 0} \frac{x(1-\cos x)}{x - \sin(x)}$ without Taylor series or L'Hôpital's rule?

I want to evaluate $\lim\limits_{x\to 0} \frac{x(1-\cos x)}{x - \sin(x)}$. We know that its plot is: And also with attention to the Taylor series we know that its series expansion at $x=0$ is: $$3 - ...
hasanghaforian's user avatar
1 vote
4 answers
95 views

Evaluating $\lim_{x\to1^+}\ln x\ln(\ln x)$ without L'hospital

I want to evaluate the following limit :$$L=\lim_{x\to1^+}\ln x\ln(\ln x)$$ My efforts: $$L =\lim_{x\to 1^+} \frac{\ln(\ln x)}{\frac{1}{\ln x}}$$ By L'hospital rule, clearly : $$L = \lim_{x\to1^+}(-\...
An_Elephant's user avatar
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3 votes
1 answer
115 views

$\lim_{x \to 0} (\cos x)^{\cot x}$

The following question is from cengage calculus . Illustration 2.95 but the explanation isn't clear to me $\lim_{x \to 0} (\cos x)^{\cot x}$ It is to be solved by using the identity : $\lim_{x \to 0} (...
Samarth Saluja's user avatar
2 votes
3 answers
116 views

How to evaluate $\,\lim\limits_{x \to 1}\left(x^n-1\right)\ln^m(1-x)\,$ without L'Hopital? [duplicate]

This question arose as part of the evaluation of $\int_{0}^{1} x^{n-1} \ln^{m}(1-x) \, \mathrm{d}x$ using integration by parts, where we would require to show that $$ L=\lim_{x \to 1} \left( x^n-1\...
Robert Lee's user avatar
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3 votes
2 answers
86 views

Can we use series expansion to evaluate $\lim_{x\to\infty}\frac{\cos x+\sin x}{x^2}?$

Evaluate $\lim_{x\to\infty}\frac{\cos x+\sin x}{x^2}$ Method $1$: Numerator is small. Denominator is much bigger. So, limit is zero. Method $2$: Using series expansion, $\lim_{x\to\infty}\frac{\left(...
aarbee's user avatar
  • 7,954
1 vote
3 answers
141 views

Evaluating $\lim_{x \to 0} \frac{x^3-x\cos x }{\sin x}$ without l'Hopital

What techniques I could use to solve this limit $$\lim_{x \to 0} \frac{x^3-x\cos x }{\sin x}$$ without l'Hopital? When I use l'Hopital the limit is $-1$. With l'Hopital $$\lim_{x \to 0} \frac{3x^2-1\...
Rosario A's user avatar
0 votes
2 answers
78 views

Show that $\lim_{x\rightarrow0}\frac{e^{\iota x}-1}{\iota x}=1$

(a) Show that $|e^{\iota a}-e^{\iota b}|\le\min\{2,|a-b|\}$ for all $a,b\in\mathbb{R}$. Hint. Write it as an integral from $a$ to $b$. (b) Show that $$\lim_{x\rightarrow0}\frac{e^{\iota x}-1}{\iota x}=...
zaira's user avatar
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0 votes
0 answers
51 views

Finding limit without L'Hopital's rule of a rational trigonometric expression

I'm having trouble evaluating the limit below, without L'Hopital's rule of a rational trigonometric expression. I've tried applying a number of trigonometric limits, but keep coming up blank $$ \lim\...
SnakeyB's user avatar
-1 votes
1 answer
79 views

How to solve this without L'Hopital? [closed]

$$f(x)=\frac{\sin(x-2)}{x^2+x-6}$$ is not defined at $x=2$ and $x=-3$, how can I solve this without using de l'Hôpital's theorem to be defined continuous at $x=2$?
Rosario A's user avatar
0 votes
2 answers
82 views

Limit of $\frac{\cos(ax)-\cos(bx)}{\cos(cx)-\cos(dx)}$ as $x$ approaches to $0$

Limit of $$\frac{\cos(ax)-\cos(bx)}{\cos(cx)-\cos(dx)}$$ as $x$ approaches to $0$. I have calculated this limit using L Hospital's rule to be $$ \frac{-a^2+b^2}{-c^2+d^2}$$ But on the book the limit ...
Dinesh Katoch's user avatar
5 votes
2 answers
308 views

Why does ${\lim_{x\to2}}\;(x-4)^x$ not exist?

Here's a question from the calculus section of AEEE 1997: $${\lim_{x\to2}}\;(x-4)^x$$ I saw this question in a very old paper of the AEEE exam ($90's$ edition of the JEE exam). So when I saw this ...
Elizabeth Huffman's user avatar
1 vote
2 answers
154 views

Evaluating $\lim_{x\to 0} \frac{ e^x - e^{-x}-2\ln(1+x)}{x \sin x}$. Why is this solution wrong?

the solution given in book is $1$ and by L'Hospital approach i do get one so what's the fault in method Question: $$\lim_{x\to 0} \frac{ e^x - e^{-x}-2\ln(1+x)}{x \sin x}$$ Work: $$\begin{...
zuck's user avatar
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1 vote
0 answers
23 views

Methods to solve limits [duplicate]

I am sort of lost when performing the limits of an equation. For instance, let's take: $\lim\limits_{x\to-\infty} x^2-7x+12$ $(-\infty)^2-7(-\infty)+12$ $(\infty)+(\infty) \rightarrow$ Thus, it is ...
Vocelia's user avatar
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2 votes
2 answers
111 views

Why can’t I divide by $n^2$ here to evaluate the limit?

$$\lim_{n\to\infty}{ \frac {n^2}{1+2+…+n}}$$ Here, if we divide and multiply by $n^2$, the numerator becomes $1$ and the limit of the denominator becomes $0$ then, the entire limit becomes infinite. ...
jsal george's user avatar
1 vote
2 answers
131 views

Can you use the Big Theorem to compute limits of rational functions at infinity?

The Big Theorem is used for sequences and natural numbers, but can we still apply it to continuous functions? Let $a_n$ and $b_n$ be sequences. $a_n \ll b_n \implies \lim_{n\to\infty} \frac{a_n}{b_n} =...
Zeal's user avatar
  • 13
2 votes
3 answers
212 views

How to check if the limit exists in multivariable calculus

$$\lim_{(x,y ) \to (0,0)} ye^{\frac{−1}{\sqrt{x^2+y^2}}}$$ I have tried many ways so far and I keep getting the limit to be $0$. So far I have set x equal to zero and then y and I got the limit to be $...
Need_MathHelp's user avatar
6 votes
4 answers
169 views

Evaluating $\lim\limits_{x\to 0}\!\left[\! \frac{1 - (\cos(2x) \cos(4x) \cos(6x) \cos(8x) \cos(10x))^3}{5 \tan^2x}\!\right]$

Following question is given in my book: Let $f(x) = \cos(2x) \cos(4x) \cos(6x) \cos(8x) \cos(10x)$ and $M = \lim\limits_{x\to 0 } \left[\dfrac{1 - f(x)^3}{5 \tan^2x}\right]$, where $M$ is finite. ...
Utkarsh's user avatar
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3 votes
2 answers
116 views

$\lim_{x\to \infty}{\bigl( \frac{(1+x)^x}{x^xe}\bigr)}^x$

$$\lim\limits_{x\to \infty}{\left( \frac{(1+x)^x}{x^xe}\right)}^x$$$$=\lim\limits_{x\to \infty}{\left( 1+\frac1x\right)}^{x^2}e^{-x}$$ Now since $\lim\limits_{x\to \infty}{\left( 1+\frac1x\right)}^{x}$...
AltercatingCurrent's user avatar
2 votes
2 answers
119 views

$\lim_{n\to \infty}(\lim_{x\to 0}( 1+\sum_{k=1}^n\sin^2(kx))^\frac{1}{n^3x^2} )$

$$\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n\sin^2(kx)\right)^\frac{1}{n^3x^2} \right)$$ $$=\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n(k^2x^2)\frac{\sin^2(kx)}{k^2x^2}\...
AltercatingCurrent's user avatar
0 votes
0 answers
27 views

Evaluate $\lim_{x\to\alpha^+}\frac{2\ln(\sqrt x-\sqrt{\alpha})}{\ln(e^{\sqrt x}-e^{\sqrt{\alpha}})}$ without L'Hopital [duplicate]

Evaluate $\lim_{x\to\alpha^+}\frac{2\ln(\sqrt x-\sqrt{\alpha})}{\ln(e^{\sqrt x}-e^{\sqrt{\alpha}})}$ without L'Hopital Using L'Hopital, I am getting $2$ as answer. I wonder how to do it without L'...
aarbee's user avatar
  • 7,954
0 votes
1 answer
33 views

Limit of function as derivative

I want to calculate $\displaystyle\lim_{x\to a}\left(\frac{f(x)}{f(a)}\right)^{\frac{1}{\ln{x}-\ln{a}}}$, where $f$ is a differentiable function at $x=a$ with $f(a)>0$. I found this exercises in ...
Benjacort's user avatar
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0 votes
3 answers
89 views

limit to infinite of a factorial number

I have this problem and I solved it like this, but muy teacer told me that it's wrong so I dont know who to do it. Prove that $\lim _{n \to \infty}\frac{(n!)^22^{2n}}{(2n)!\sqrt{n}}$. I use the ...
Mingyu's user avatar
  • 1
2 votes
1 answer
70 views

What is $\,\lim\limits_{x\to0}\frac{\ln\left(\frac{e^x}{x+1}\right)}{x^2}\;?$

I've been trying to solve this limit, but can't really seem to be able to without using l'Hospital's rule (which the textbook specifically forbids). Here goes: $$\lim_{x\to0}\frac{\ln\left(\frac{e^x}{...
Heribert Greinix's user avatar

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