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?
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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:
$$\lim_{(x,y)\to(0,0)}\frac{x^2y^3\...
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This is a problem from Differential Calculus for Beginners by Joseph Edwards [duplicate]
$$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]$$
Can ...
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If $|f'(x)-e^{2x}|\leq 3$ for all $x\in\mathbb{R}$. Then evaluate the limit $\lim_{x\to +\infty}\frac{f(x)}{e^{2x}}$
Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that $f'$ is continuous and satisfies $|f'(x)-e^{2x}|\leq 3$ for all $x\in\mathbb{R}$. Then evaluate the limit $$\lim_{x\to +\infty}\...
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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 ...
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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!...
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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 ...
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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?
$$\lim_{x\to 0} \left(\frac{1}{\sin^2(x)} + \frac{\cos^2(x)}{...
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Evaluating $\lim_{x\to\infty}\frac{\ln\cos\frac {48} x}{\ln\cos\frac{1}{12x}}$ without using L'Hopital
I've seen some similar question, but none with $x\to\infty$
$$\lim_{x\to\infty}\frac{\ln\cos\frac {48} x}{\ln\cos\frac{1}{12x}}$$
one of the ideas that i tried is, with $t=\frac{48}{x}$, changing the ...
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Evaluating $\lim_{x\to0}\frac{\sqrt{\frac{1}{\cos x}}-1}{\sin^2 {\frac{x}{16}}}$ without L'Hopital's Rule
I was solving the limit
$$\lim_{x\to0}\frac{\sqrt{\frac{1}{\cos x}}-1}{\sin^2 {\frac{x}{16}}}$$
I simplified the numerator by multiplying both the numerator and denominator by $\sqrt{\frac{1}{\cos x}}+...
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Why doesn't L'Hôpital's Rule work on $\lim_{T\to\infty} \frac{\frac12T-\frac{1}{4}\sin2T}{T}$? [duplicate]
$$\lim_{T\to\infty} \frac{\frac{T}{2}-\frac{1}{4}\sin2T}{T}$$
If I solve this limit by breaking it into 2 parts and consider $\lim_{T\to\infty} \frac{sin2T}{T}=0$ and $\lim_{T\to\infty}\frac{T/2}{T}=\...
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What is infinite limit of rational trigonometric function? [closed]
How can we justify the following limit?
$$
\lim _{x \rightarrow -\infty} \dfrac{1+\cos x}{3+2\cos x}
$$
We can see the result of limit DNE or $[0, 2]$ on the graph. How can we show?
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show that $\lim_{x\rightarrow-\infty}\frac{1}{x^k}=0$
I am reading Calculus, Purcell and there is a solved example
Show that if $k$ a positive integer, then
$$\lim_{x\rightarrow\infty}\frac{1}{x^k}=0\ \text{ and} \ \lim_{x\rightarrow-\infty}\frac{1}{x^k}=...
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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'...
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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\...
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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. ...
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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 ...
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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 $...
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2
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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 ...
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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}|...
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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(\...
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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
$$
\lim_{y \to 0^+}y^β(-\log_a y)^α = 0.\tag{*}\label{*...
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Two possible answers for α and β
So there was this limits question in the JEE Main 2022 exam:
If $\lim\limits_{n \to \infty}( \sqrt{n^{2}-n-1}+n\alpha +\beta)=0$
Find $8(\alpha + \beta)$
It had the following options:
A. 4
B. -8
C. 8
...
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How to solve $\lim\limits_{x\to 1}\frac{\sin \pi x}{x-1}$ without L'Hospital [closed]
$\lim\limits_{x\to 1}\frac{\sin \pi x}{x-1}$
Been struggling solving this without L'Hospital
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Limit of $x[3−\cos(x^2)]$ using the epsilon-delta definition
This is a duplicate of this post here, but I'm new to Stack Exchange so I'm not able to comment on that post yet (and sorry in advance if this question isn't appropriate to the platform).
I was ...
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trigonometric limit factoring
I have this problem that I can't solve:
$$\lim_{x \to 0^+} \frac{3x-\sin 2x}{\sqrt{1-\cos x}}$$
Tried to multiply by the conjugate, so I can get rid of the squareroot on the denominator, but I am ...
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Evaluate $ \lim_{n \to \infty} \sin^n\left(\frac{2\pi n}{3n+1}\right)$
I'm trying to solve this limit rigorously
$ \lim_{n \to \infty} \sin^n\left(\frac{2\pi n}{3n+1}\right)$
I can see that the answer is $0$ as
$$\lim_{n \to \infty} \sin\left(\frac{2\pi n}{3n+1}\right) ...
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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:
$$3 - ...
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Evaluating $\lim_{x\to1^+}\ln x\ln(\ln x)$ without L'hospital
I want to evaluate the following limit :$$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^+}(-\...
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$\lim_{x \to 0} (\cos x)^{\cot x}$
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} (...
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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
$$
L=\lim_{x \to 1} \left( x^n-1\...
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Can we use series expansion to evaluate $\lim_{x\to\infty}\frac{\cos x+\sin x}{x^2}?$
Evaluate $\lim_{x\to\infty}\frac{\cos x+\sin x}{x^2}$
Method $1$:
Numerator is small. Denominator is much bigger. So, limit is zero.
Method $2$:
Using series expansion,
$\lim_{x\to\infty}\frac{\left(...
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Evaluating $\lim_{x \to 0} \frac{x^3-x\cos x }{\sin x}$ without l'Hopital
What techniques I could use to solve this limit
$$\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\...
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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 $$\lim_{x\rightarrow0}\frac{e^{\iota x}-1}{\iota x}=...
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Finding limit without L'Hopital's rule of a rational trigonometric expression
I'm having trouble evaluating the limit below, without L'Hopital's rule of a rational trigonometric expression.
I've tried applying a number of trigonometric limits, but keep coming up blank
$$ \lim\...
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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 $x=2$?
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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 ...
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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 ...
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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:
$$\begin{...
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Methods to solve limits [duplicate]
I am sort of lost when performing the limits of an equation.
For instance, let's take: $\lim\limits_{x\to-\infty} x^2-7x+12$
$(-\infty)^2-7(-\infty)+12$
$(\infty)+(\infty) \rightarrow$ Thus, it is ...
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Why can’t I divide by $n^2$ here to evaluate the limit?
$$\lim_{n\to\infty}{ \frac {n^2}{1+2+…+n}}$$
Here, if we divide and multiply by $n^2$, the numerator becomes $1$ and the limit of the denominator becomes $0$ then, the entire limit becomes infinite. ...
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Can you use the Big Theorem to compute limits of rational functions at infinity?
The Big Theorem is used for sequences and natural numbers, but can we still apply it to continuous functions?
Let $a_n$ and $b_n$ be sequences. $a_n \ll b_n \implies \lim_{n\to\infty} \frac{a_n}{b_n} =...
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How to check if the limit exists in multivariable calculus
$$\lim_{(x,y ) \to (0,0)} ye^{\frac{−1}{\sqrt{x^2+y^2}}}$$
I have tried many ways so far and I keep getting the limit to be $0$. So far I have set x equal to zero and then y and I got the limit to be $...
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Evaluating $\lim\limits_{x\to 0}\!\left[\! \frac{1 - (\cos(2x) \cos(4x) \cos(6x) \cos(8x) \cos(10x))^3}{5 \tan^2x}\!\right]$
Following question is given in my book:
Let $f(x) = \cos(2x) \cos(4x) \cos(6x) \cos(8x) \cos(10x)$ and $M = \lim\limits_{x\to 0 } \left[\dfrac{1 - f(x)^3}{5 \tan^2x}\right]$, where $M$ is finite.
...
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$\lim_{x\to \infty}{\bigl( \frac{(1+x)^x}{x^xe}\bigr)}^x$
$$\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 $\lim\limits_{x\to \infty}{\left( 1+\frac1x\right)}^{x}$...
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$\lim_{n\to \infty}(\lim_{x\to 0}( 1+\sum_{k=1}^n\sin^2(kx))^\frac{1}{n^3x^2} )$
$$\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n\sin^2(kx)\right)^\frac{1}{n^3x^2} \right)$$
$$=\lim_{n\to \infty}\left(\lim_{x\to 0}\left( 1+\sum_{k=1}^n(k^2x^2)\frac{\sin^2(kx)}{k^2x^2}\...
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Evaluate $\lim_{x\to\alpha^+}\frac{2\ln(\sqrt x-\sqrt{\alpha})}{\ln(e^{\sqrt x}-e^{\sqrt{\alpha}})}$ without L'Hopital [duplicate]
Evaluate $\lim_{x\to\alpha^+}\frac{2\ln(\sqrt x-\sqrt{\alpha})}{\ln(e^{\sqrt x}-e^{\sqrt{\alpha}})}$ without L'Hopital
Using L'Hopital, I am getting $2$ as answer.
I wonder how to do it without L'...
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Limit of function as derivative
I want to calculate $\displaystyle\lim_{x\to a}\left(\frac{f(x)}{f(a)}\right)^{\frac{1}{\ln{x}-\ln{a}}}$, where $f$ is a differentiable function at $x=a$ with $f(a)>0$.
I found this exercises in ...
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limit to infinite of a factorial number
I have this problem and I solved it like this, but muy teacer told me that it's wrong so I dont know who to do it.
Prove that $\lim _{n \to \infty}\frac{(n!)^22^{2n}}{(2n)!\sqrt{n}}$.
I use the ...
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What is $\,\lim\limits_{x\to0}\frac{\ln\left(\frac{e^x}{x+1}\right)}{x^2}\;?$
I've been trying to solve this limit, but can't really seem to be able to without using l'Hospital's rule (which the textbook specifically forbids). Here goes: $$\lim_{x\to0}\frac{\ln\left(\frac{e^x}{...
