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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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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35 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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171 views

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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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
43 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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19 views

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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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 ...
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2answers
71 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
26 views

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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29 views

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
719 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
84 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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424 views

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
60 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 ...
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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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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? ...
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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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27 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
311 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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97 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 ...
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49 views

why is $\displaystyle \frac{\log(\sin x)}{\log(x)}$ $\quad\frac{\infty}{\infty}$ form as $x\to 0$?

In this question $\displaystyle\frac{\log(\sin x)}{\log x}$ is taken as $\displaystyle\frac{\infty}{\infty}$ indeterminate form. But $\log(0)$ is not defined so how can L'Hospital's rule can be ...
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1answer
57 views

Why is this rational expressions indeterminate when evaluated?

I have this rational expression to evaluate, $$ {{3a-3}\over {4a(a-1)}} \text { if } a=1. $$ I understand that if you substitute 1, both the numerator and denominator would turn out 0, thus making ...
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computing a limit of a function that is positive defined

given a function $f(x)$ positive and continuous at $x=a$, with $f(a)\ne0$, compute: $$\lim_{n\to+\infty}\left[\frac{f\left(a+\frac{1}{n}\right)}{f(a)}\right]^n$$ a friend asked me that, though I ...
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indeterminate limit where applying L'Hopitals Rules directly doesn't help and using ln gives wrong answer

I am trying to determine the limit $\displaystyle{ \lim_{x \to 0^-}{\frac{-e^{1/x}}{x}}}$. Plugging in $x$ directly, yields $0/0$ which is indeterminate. Applying L'Hopitals rule does not simplify ...
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I got the answer for $\lim \limits_{x \to \infty} {\left({3x-2 \over3x+4}\right)}^{3x+1}$, but only by a mistake - how do I solve correctly?

This is what I did for: $$\lim \limits_{x \to \infty} {\left({3x-2 \over3x+4}\right)}^{3x+1}$$ Check form: $\left({\infty \over \infty}\right)^{\infty}$. Apply L'Hospital's Rule to just $\lim ...
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2answers
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Easier way to solve $\lim \limits_{x \to 0} \space \cot{x}-{1 \over x}$ using L'Hospital's Rule?

This is what I did for: $$\lim \limits_{x \to 0} \space \cot{x}-{1 \over x}$$ Check form: $\infty - \infty$. Rearrange it to be a quotient: $$\begin{align} \\ & =\lim \limits_{x \to 0} \space ...
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Is $(-1)^{\infty}$ an indeterminate form?

We know that $\lim_{n\to\infty}(-1)^n=(-1)^{\infty}$ doesn't exist. Now take $\lim_{n\to\infty}(-1)^{2n}=(-1)^{\infty}$. This limit exists, because ...
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112 views

Determination of $1^\infty$ indeterminate forms

Recently I have been learning some of the basic concepts of limits and in my academics. There I have been taught some methods to evaluate indeterminant forms like $1^\infty$, $0^0$ and $\infty^0$. ...
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How to explain indeterminations, and some aprpoaches to $+\infty$ or $-\infty$, for middle school students?

Question: how to explain the undefinitions $0^0$ and $\frac{0}{0}$ for Middle school students?? I am a math teacher and I don't know how to answer properly when studens ask me why some operations ...
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Using L'Hospital's Rule to evaluate limit to infinity

I'm given this problem and I'm not sure how to solve it. I was only ever given one example in class on using L'Hospital's rule like this, but it is very different from this particular problem. Can ...
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1answer
78 views

Help me integrate this function using Simpson's rule

I have a question: compute $$\int_0^1 \frac{\sin(x)}{x}\,dx$$ for $n=10$ divisions. I got the value $0.9127$ but I think its a bit too high.
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Why can $2^3$ be defined but $0^0$ cannot

From what I gather, we can't just define $0^0$ to be $0$ or $1$ or $69$ or whatever, because $\lim\limits_{x\mathop\to0}0^x=0$ and $\lim\limits_{x\mathop\to0}x^0=1$. So $0^0$ is called indeterminate ...
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1answer
315 views

Limits of Indeterminate Powers in Exponential Form using L'Hopital's Rule

I am trying to find the limit as $x \rightarrow 0$ of $x^x$ using L'Hopital's rule. I have written it in exponential form: $\lim\limits_{x \rightarrow 0} e^{x \ln x}$. I do not know how to put it in ...
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Limit of a polynomic-exponential sequence

I have to calculate the following limit: $$L=\lim \limits_{n \to \infty} -(n-n^{n/(1+n)})$$ I get the indeterminate form $\infty - \infty$ and I don't know how to follow. Any idea? Thank you very ...
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Indeterminate limit that is supposed to be solved with De L'Hospital's rule

Last week my Maths teacher gave the class this exercise taken from our text book. We are working on De L'Hospital's rule at the moment and this exercise is from that part of the book so everybody ...
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Is zero a singular point of this function?

$$f(z)=\frac{z^3}{z+z^5}$$I thought that this function has 5 singular points. But my friend is convinced it only has four because if you write is as$$f(z)=\frac{z^2}{1+z^4}$$ then it is defined at ...
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Solving limit without L'Hopital

I'd like some help in solving this limit without using L'Hopital. $$\lim_{x\to -\infty}\frac{\ln(1-2x)}{1-\sqrt{1-x}}$$ I've also solved it changing the variable to $y=\sqrt{1-x}$ but I would like ...
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3answers
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Is $\frac{0}{\infty}$ indeterminate?

I have been searching for an answer for this for half an hour and I can't seem to find one. I've lots of information about other combinations of $0$ and $\infty$ but I haven't seen anything that says ...
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Is $0^\infty$ indeterminate?

Is a constant raised to the power of infinity indeterminate? I am just curious. Say, for instance, is $0^\infty$ indeterminate? Or is it only 1 raised to the infinity that is?
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How do I find this limit: $\lim_{x \to \infty} \sqrt{x^4-3x^2-1}-x^2$ [duplicate]

$$ \lim_{x \to \infty} \sqrt{x^4-3x^2-1}-x^2 $$ The answer is $$ \frac{-3}{2} $$ according to Wolfram alpha.
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Is it wrong to tell children that $1/0 =$ NaN is incorrect, and should be $∞$?

I was on the tube and overheard a dad questioning his kids about maths. The children were probably about 11 or 12 years old. After several more mundane questions he asked his daughter what $1/0$ ...
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3answers
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$\lim_{x \to \infty} \sqrt{4x^2 + 4} - (2x + 2)$ [duplicate]

Possible Duplicate: Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$ $$\lim_{x \to \infty} \sqrt{4x^2 + 4} - (2x + 2)$$ So, I have ...
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Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$

What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty} \sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x.$$ In other words, if I am given a polynomial $P(x)=x^n + a_{n-1}x^{n-1} ...
73
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Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are ...