If the expression obtained after any substitution during limit analysis does not give enough information to determine the original limit, it is known as an indeterminate form.

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Indeterminate form and equality of expressions.

Given this chain of equalities: $$\frac{\sqrt{1-x} - \sqrt{1+x}}{x}=\frac{(\sqrt{1-x} - \sqrt{1+x})(\sqrt{1-x} + \sqrt{1+x})}{x(\sqrt{1-x} + \sqrt{1+x})} = \frac{(\sqrt{1-x})^2 - ...
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+50

Existence and non-singularity of the Fisher information matrix

Consider a random vector $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X: \Omega \rightarrow \mathbb{R}^k$. Suppose $X$ has probability density $p_{\theta_0}$ with respect ...
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Limit $\lim_{x \to 0} \dfrac{\sin(1-\cos x)}{ x} $

$$\lim_{x \to 0} \dfrac{\sin(1-\cos x)}{ x} $$ What is wrong with this argument: as $x$ approaches zero, both $x$ and $(1-\cos x)$ approaches $0$. So the limit is $1$ . How can we prove that they ...
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6answers
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“Proving” that $0^0 = 1$ [duplicate]

I know that $0^0$ is one of the seven common indeterminate forms of limits, and I found on wikipedia two very simple examples in which one limit equates to 1, and the other to 0. I also saw here: ...
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23 views

Does symmetric property of Equivalence hold when the left hand side of the equation is in indeterminate form?

The Symmetric Property of Equivalence is a=b implies b=a. This property does not have any conditions on a or b. But what if through manipulation I get $a=\frac00$ where $a$ is a real number does this ...
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2answers
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Need help solving these limits [closed]

$$1. \lim_{n \to \infty} \left(\frac{2n^2 + 5}{1 + 5 + ... + (4n - 3)} + \frac{5}{n}\right)$$ $$2. \lim_{x \to -1} \left(\frac{x^4 + 2x^3 + 4x^2 + 6x + 3}{x^3 - 3x^2 - 9x - 5}\right)$$ $$3. \lim_{x ...
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List of Indeterminate forms in Mathematics

I know that , $1) \frac{0}{0}$ $2) \frac{\pm\infty}{\pm\infty}$ $4) 0 \times(\pm\infty) $ are Indeterminate forms. But in measure theory $ 0 \times(\pm\infty) =0 $ Are there any other ...
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284 views

Does $1^{\infty}=e$ or $1^{\infty}=1$?

In fact the real question is: Does $\lim\limits_{n\to\infty}1^{n}=e$?. I know that $$ \lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n=e, $$ So, can we say that $1^\infty=e$? And, by logic, ...
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Plotting a function around $0$ shows it is jumping around, although the limit as $x\to 0$ exists

My problem revolves around the function: $$ f(x) = \frac{\sin(\tan x) - \tan(\sin x)} {\arcsin(\arctan x) - \arctan(\arcsin x)} $$ The $\lim_{x\to 0}f(x)= 1$. However, whilst approaching 0, there is ...
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49 views

How do you prove indeterminate form using epsilon and delta?

The question, for instance, is proving $$\lim_{x\to\infty}\frac{x}{x+1}=1$$ This is my answer, which is likely incorrect. $$\forall\epsilon>0, \exists M \in \mathbb{R}$$ such that $$ x>M ...
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3answers
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Evaluating the limit $\mathop {\lim }\limits_{x \to 0} \frac{{x(1 - 0.5\cos x) - 0.5\sin x}}{{{x^3}}}$

For evaluating the limit $\lim\limits_{x \to 0} \frac{x(1 - 0.5\cos x) - 0.5\sin x}{x^3}$, I proceeded as follows: $$\lim_{x \to 0} \left(\frac{x(1 - 0.5\cos x)}{x^3} - ...
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0answers
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How to solve a particular indeterminate form

So the answer says $$\lim_{x\to \infty}x^2\sin\left(\frac1x\right)=\lim_{h\to 0^+}\frac1h\frac{\sin h}h$$ How does the transformation work?
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Question about the derivative definition

The derivative at a point $x$ is defined as: $\lim\limits_{h\to0} \frac{f(x+h) - f(x)}h$ But if $h\to0$, wouldn't that mean: $\frac{f(x+0) - f(x)}0 = \frac0{0}$ which is undefined?
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What's the value of $\frac{(x-1)x^2}{x-1}$ when $x=1$?

Consider the function $$f(x)= \frac{(x-1)x^2}{x-1},\quad x \in \Bbb R\ .$$ This function gives simply gives $f(x)=x^2$ by cancelling the term $x-1$, if I am not wrong. The variable here is $x$ which ...
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2answers
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Prove that $\lim_{x\to 0}\frac{1}{\ln(1+x)}-\frac{1}{\ln\left(x+\sqrt{1+x^2}\right)}=\frac{1}{2}$ [duplicate]

Prove that $$\lim_{x\to 0}\frac{1}{\ln(1+x)}-\frac{1}{\ln\left(x+\sqrt{1+x^2}\right)}=\frac{1}{2}$$ I can solve this problem using L Hospital Rule by converting it into $\frac{0}{0}$ form. But the ...
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5answers
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How do I evaluate this limit :$\displaystyle \lim_{x \to \infty} ({x\sin \frac{1}{x} })^{1-x}$.?

I would like to evaluate this limit :$$\displaystyle \lim_{x \to \infty} ({x\sin \frac{1}{x} })^{1-x}$$. I used taylor expansion at $y=0$ , where $x$ go to $\infty$ i accrossed this problem : ...
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Indeterminate forms of Limits in Multivariable Calculus

In multivariable calculus, I have come across different approaches to operate on indeterminate forms. I have come across cases where you choose different paths to test whether the limit takes the same ...
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5answers
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How do I solve $\lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1-x^2}}{\sqrt{1+x}-1}$ indeterminate limit without the L'hospital rule?

I've been trying to solve this limit without L'Hospital's rule because I don't know how to use derivates yet. So I tried rationalizing the denominator and numerator but it didn't work. $\lim_{x \to ...
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If 0 / 0 is indeterminate, are all clauses “0 / 0 != x” true

Elsewhere arose a discussion about logical clauses that can be made from indeterminate forms, in this case, namely $0 / 0$. Since $0 / 0$ is indeterminate form, can we make these logical clauses: $0 ...
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Having trouble with certain indefinite forms of a limit

I have been having a rather difficult time trying to solve this limit $$ \lim_{x\to0}\bigg(\frac{5}{x^4}-\frac{5}{x^2}\bigg) $$ So far, I have rewritten it to this point $$ ...
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Is $\lim\limits_{x,y\to-\infty}\frac{\sqrt x\sqrt y}{\sqrt{xy}}=1$?

WolframAlpha is suggesting (judging by the plot given) that the limit is actually $-1$. I would think the following manipulations would be okay to conclude that is the opposite. ...
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1answer
26 views

Test for convergence - Log operation

∑ Log (n / n+ 1) I solved the above problem in two different ways. 1st Method = Log n - Log n+1 = Log n - Log n . Log 1 = Log n - Log n . 0 = Log n = Log ∞ = ∞ (Diverges) Method 2 = Log ( n / n + ...
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2answers
39 views

Prove that $\lim_\limits{x\to 1}{\frac{f^2(x)+g^2(x)}{f(x)+g(x)}}=+\infty$

Let $f,g:\mathbb{R}\rightarrow\mathbb{R^*_+}$ be functions such that: $\lim_\limits{x\to 1}{f(x)}=+\infty$ and $\lim_\limits{x\to 1}{g(x)}=+\infty$ Prove that: $$\lim_\limits{x\to ...
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2answers
71 views

The indeterminacy of 0/0 and vacuous truth?

Today, my roommate and I picked up our friend from the airport. We were supposed to pick him up yesterday, but he missed his flight. We joked that he misses flights a lot, and that only catches 70% of ...
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1answer
37 views

Conventions adopted for extended reals

It is known that $0^0$ despite being an indeterminate limit form, is usually defined to be equal to $1$. I wonder whether similar conventions exist for some other "indeterminate forms" in the context ...
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Why does L'Hôpital's rule work for sequences?

Say, for the classic example, $\frac{\log(n)}{n}$, this sequence converges to zero, from applying L'Hôpital's rule. Why does it work in the discrete setting, when the rule is about differentiable ...
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2answers
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Is $\frac00=\infty$? And what is $\frac10$? Are they same? Does it hold true for any constant $a$ in $\frac{a}0$ [duplicate]

I know that $\lim_{x\to0}\frac{x}{x}=$ 1. But in my text book, it is written that it is $\infty$ and even $\frac10=\infty$. But how is it possible? And are they both same? What is the difference ...
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3answers
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Why don't we use undefined forms to find limits of infinity? [closed]

I came across a couple of simple limits of infinity and solved them using the normal subsequence method. Yet, I also tried solving them by substituting the undefined form $\frac{1}{0}$, and although ...
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1answer
46 views

What does it mean for a “formula to be undefined”?

I was covering the techniques used sketch rational functions of five different types as follows: However, then I encountered this: And, Ij just can't find out what it means for the formula to ...
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Infinity indeterminate form that L'Hopital's Rule: $\lim_{x\to0^+}\frac{e^{-\frac{1}{x}}}{x^{2}}$

When I tried to find the limit of $$ \lim_{x\to0^+}\frac{e^{-\frac{1}{x}}}{x^{2}} $$ by applying L'Hopital's Rule the order of denominator would increase. What else can I do for it?
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4answers
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Undefined reference $1^{\infty}$ in a limit such as $\lim_{n\to\infty}(1+1^n)$ [closed]

If $1^{\infty}$ is undefined reference in a limit, how is $$\lim\limits_{n\to\infty}(1+1^n)=2$$
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1answer
47 views

Indeterminant in Summation

I have the following summation: $$\sum_{k=0}^\infty(1-e^x)^k=\sum_{k=0}^\infty\sum_{j=0}^k\binom{k}{j}(-1)^je^{jx}$$ Then $$e^{jx}=\sum_{i=0}^\infty j^i\frac{x^i}{i!}$$ So, ...
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1answer
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If period of $\sin^{2m} \sqrt{k}x$, $m\in \mathbb N$ is $\pi$ then find $\lim \limits_{n\to \infty }k^{n}$.

If period of $\sin^{2m} \sqrt{k}x$, $m\in \mathbb{N}$ is $\pi$ then find $\lim \limits_{n\to \infty }k^{n}$. My attempt: Period of $$\sin^{2m} \sqrt{k}x=\frac{\pi}{\sqrt{k}}$$ According to ...
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3answers
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How to solve this particular indetermination: $0*\infty$

The limit in question is: $$\lim_{\color{red}n\to\infty} 2n\left(\sqrt{n^6+5n^2}-n^3\right)$$ By looking it up on wolfram alpha I found out the answer is 5 but I am not so sure how to arrive to it. I ...
3
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2answers
87 views

Evalutating $\lim_{x\to +\infty} \sqrt{x^2+4x+1} -x$ [duplicate]

I'm looking to evaluate $$\lim_{x\to +\infty} \sqrt{x^2+4x+1} -x$$ The answer in the book is $2$. How do I simply evaluate this problem? I usually solve limits such as this with the short cut ...
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1answer
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Difference b/w Undefined and Indeterminate quantities. Can two of them be equaled to each other?

While studying limits (continuity of functions to be specific), I encountered with two indeterminate forms for LHL and RHL. Both being $\color{blue}{\frac{1}{0}}$ . As per what I've read over ...
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limits as $x\rightarrow\pm\infty$ of indeterminate forms $\frac{a^x+b^x}{c^x+d^x}$, where $a,b,c,d\in\mathbb{R}$

Good day sirs would you kindly help me to find the limit of $\frac{a^x+b^x}{c^x+d^x}$ as $x\rightarrow\pm\infty$, where $a$,$b$,$c$ and $d$ are real numbers? I already know how to use the L' ...
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5answers
740 views

My dilemma about $0^0$ [duplicate]

We know that $0^0$ is indeterminate. But if do this: $$(1+x)^n=(0+(1+x))^n=C(n,0)\cdot ((0)^0)((1+x)^n) + \cdots$$ we get $$(1+x)^n=(0^0)\cdot(1+x)^n$$ So, $0^0$ must be equal to $1$. What is ...
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Limit of an indeterminated form?

I want to find: $$\lim_{t\to\infty}X_{t}$$ where: $$X_{t} = \frac{A_{t}^{a}}{B_{t}}$$ I know that: $$\lim_{t\to\infty}A_{t}=0$$ and $$\lim_{t\to\infty}B_{t}=0$$ Can I say with certainty that: ...
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1answer
116 views

What is 0 raised to 0 ???!!!! [duplicate]

I have read many articles on this confusion but i am still confused... My simple question is - What is $0^0$? What is the present agreement to this? I feel that it should be 1 as anything to ...
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3answers
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Question on using L'Hopital's rule for this problem?

I'm doing some practice questions and I've encountered a wall. The question is: Find the limit of the function $(\ln4x-\ln(x+7))$ as $x \rightarrow \infty$. the indeterminate form is ...
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Determining the indeterminate limit of an exponent problem

I have the equation: $y=\left( 1+\frac{1}{2}^{x} \right)^{x}$ In evaluating it's limit as it approaches +$\infty$, I can't seem to simplify the expression to a non-indeterminate form. By graphing I ...
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Usual and “unusual” indeterminate forms

I've always read there are seven "main" indeterminate forms, 0/0, ∞/∞,∞−∞,0∞,00,1∞,∞0. I've recently started working more keenly on analysis, and the seven "main" forms are never expanded to include ...
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3answers
88 views

Need help with a limit to infinity involving a radical with indeterminate form (stuck in the factoring)

this is my first time on Math Exchange, I searched around the site and could not find a question for this math problem so I do not believe that I am asking a previously asked question, if I am please ...
3
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3answers
75 views

How to evaluate the limit $\lim_\limits{x\to 0+ } \frac{1}{\sqrt{x}}\left ( \frac{1}{\sin x} - \frac{1}{x}\right )$?

$$\lim_{x\to 0^+ } \frac{1}{\sqrt{x}}\left ( \frac{1}{\sin x} - \frac{1}{x}\right ) =\ ?$$ I rearranged it as $$\lim_{x\to 0^+ } \frac{x-\sin x}{x\sqrt{x}\sin x} = \lim_{x\to0^+ } \frac{x-\sin ...
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2answers
59 views

Indeterminate form as a series

We know that $0 \times \infty$ is an indeterminate form. However, is it equivalent to $0 + 0 + 0 + \cdots$? If yes, why we do not consider $\displaystyle \sum_{n = 0}^\infty 0$ an indeterminate form? ...
2
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1answer
37 views

computing an indeterminate form of a limit

I need a hint on computing this one limit: $$\lim_{n \to\infty}\,\frac{2\cdot4^n+3\cdot n^4}{4\cdot n^6-3\cdot 3^n +3n}$$ Thank you
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1answer
31 views

Limit Problem : Infinity Limit Problem

I can't find the way to solve this question and i always get 0/0. The question is: $\lim_{x\rightarrow \infty } x\left [ 2^{\frac{1}{x}}-1 \right ]$ From Mathematica, i get -infinity. But how can it ...
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1answer
1k views

What is the value of 0/0? [duplicate]

I have heard that anything divided by zero is infinity, so i was wondering what would be 0/0? I am a high school student and I haven't studied calculus yet.
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2answers
112 views

Why Is $1^{\infty}$ an indeteminate form? [duplicate]

For me, multiplication is a binary operation so it can be applied only on a finite sequence of numbers. but $1^{\infty}$ requires that we apply multiplication infinitly which is not defined as ...