Questions tagged [limits-without-lhopital]
The evaluation of limits without the usage of L'Hôpital's rule.
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Computing the Limit $\lim_{n \to \infty} n\left[ \frac{a_{n+1}}{a_{n}} - \left(\frac{n}{n+1}\right)^{\frac{1}{3}} \right]$
Problem Statement:
Let $\displaystyle a_{n} = \frac{\left(2n\right)!}{\left(n!\right)^{2}4^{n}}$.
$$
\mbox{Compute the limit:}\quad
\lim_{n \to \infty}\left\{n\left[\frac{a_{n + 1}}{a_{n}} - \left(\...
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Proving $\lim_{y\to 0} y^2 \ln|yx^2|=0$ using sequences
I have the function
$$f(x, y) = y^2 \ln|yx^2|$$ and I want to prove that $f(x, 0)$ goes to zero but using sequences.
SO I thought about this: I choose $b_n = \frac{1}{n}$, and in general any $b_n$ ...
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Evaluate $\lim_{x \to π/4} \frac{\sqrt{2}- \cos(x)- \sin(x)}{(4x-π)^2}$ [closed]
$$\displaystyle \lim_{x \to π/4} \frac{\sqrt{2}- \cos(x)- \sin(x)}{(4x-π)^2}$$
Can anyone pls help me to figure out the solution without using L'Hopital's rule as I thought if there was an alternative ...
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Solving a combined limit with an $1^{\infty}$ form nested inside a 0×∞ form
I came across this limit problem:
$\lim _{x \rightarrow \infty}\left\{\left(\frac{x+1}{x-1}\right)^x-e^2\right\} \cdot x^2$
Plugging this into desmos, one can see that the limit approaches $\frac{2 e^...
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Do you need L'Hôpital's rule to prove Taylor's formula?
I recently read a Quora answer. The answerer was asked to solve the limit
$$\lim_{x\to0}\frac{\cos x-e^x}{\sin x}$$
without using L'Hôpital's rule. The answerer used the Taylor series expansion of the ...
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Is it possible to show that for $q>0$, $\lim\limits_{x\to\infty}\dfrac{(\ln{x)^p}}{x^q} = 0$ without using L'Hopital's Rule?
Is it possible to show that for $q>0$, $\lim\limits_{x\to\infty}\dfrac{(\ln{x)^p}}{x^q} = 0$ without using L'Hopital's Rule?
Applying L'Hopital's Rule repeatedly until the numerator becomes a ...
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Limit of square root, not working as expected [duplicate]
Hey i have this function, and I don't understand why I get wrong limit if I insert x into the square root, even though it's correct algebraic to insert it.
$$
\frac{\sqrt{x^2 + 9}}{x}
$$
The first ...
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Show that $(1-\frac{1}{n})^{n^2}$ converges to $0$ [duplicate]
I want to show that
$\displaystyle\quad\lim_{n \to \infty}\left(1 - {1 \over n}\right)^{n^{2}} = {\large 0}$
This is in a context where L'Hopital isn't allowed, but is known that
$$
{\rm e}^{\large a} ...
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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:
$$\lim_{x\to ∞...
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Ratio of two diverging integrals
Consider the ratio:
$$
r = \frac{\displaystyle\int_{-\infty}^{\infty}dx\, e^{-x^2 / 2a} x^2}{\displaystyle\int_{-\infty}^{\infty}dx\, e^{-x^2 / 2a}}
$$
For $a > 0$ we have $r = a$ because after ...
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Question Regarding I.F. of limits
My calculus class has already gone over indeterminate forms, l'Hospital rule, etc. and I am preparing for an exam. One thing I don't understand in my professor's notes, is a part of the following sum ...
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Continuous differentiability of an exponential function at $x=y=0$
I have a function, $f(x,y) = \sqrt{x^2 + y^2} \exp(-\sqrt{x^2 + y^2}) ~\forall (x,y) \in \mathbb{R}^2$. I have been trying to check whether this function is continuously differentiable (or has bounded ...
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Finding the limit using sandwich
I got stuck finding :
$$\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}=\...
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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)...
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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+...
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