Questions tagged [indeterminate-forms]

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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1answer
59 views

Is this regarding $0^0$ correct?

Recently, I was working on numbers and I accidentally discovered the below proof : $$^nx=e^{e^{^{n-2}x\ln x+\ln(\ln x)}}$$ Also, we know that $$e^z=\sum\limits_{N=0}^{\infty}\frac{z^N}{N!}; \ ^{-1}x=0$...
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37 views

calculating $\lim\limits _{x\to0}\left(\frac{1}{x^{2}}-\frac{e^{x}}{\left(e^{x}-1\right)^{2}}\right)$ using taylor polynomials

I am trying to calculate the limit $$\lim\limits _{x\to0}\left(\frac{1}{x^{2}}-\frac{e^{x}}{\left(e^{x}-1\right)^{2}}\right)$$ using taylor polynomials. I tried using L'Hôpital's rule, but it was ...
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21 views

Indeterminate form in PDF

In the paper On the distribution of the product of correlated normal random variables I found an analytical formula for PDF of an averaged multiplication of two random variables: $$f(z) = \frac{n^{(n+...
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2answers
77 views

Why is $0/\lim_{x\to0} x$ undefined, when $\lim_{x\to2} (x^2-4)/(x-2)$ is defined?

For $\lim\limits_{x\to2}\ (x^2-4)/(x-2)$, we are able to cancel out $x-2$ and rewrite it as $\lim\limits_{x\to2} x+2 $ But in maths, we are not able to cancel out $0$ values so $x-2$ is not zero, and ...
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60 views

$0^\infty$ and indeterminate forms

I have actually two related questions: 1. Should indeterminate forms be able to attain an infinite number of values to be considered indeterminate? I’m asking because Wikipedia says: The expression 1/...
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1answer
32 views

How to prove that a form like $0^\infty$ is not an indeterminate form? What makes a form indeterminate?

I just wanted to know why some particular forms (seven of them) are called indeterminate forms. Why are there only seven indeterminate forms? Can anyone please prove why $0^\infty$ is not an ...
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15 views

writing a vector using null space and range of a matrix.

for a non-zero matrix A and non-zero vector b , imagine $b=b_R+b_N$ which $b_R\in range (A)$ and $b_N\in null (A^T)$. 1-prove that $b_R$ and $b_N$ are unique. 2-prove that $b^T_Rb_N=0$. 3-prove that ...
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23 views

Mathematical notation in indeterminate limits

I aspire to be a mathematician one day and I'm trying my best to perfect my form and my notation in mathematical writing. Therefore I often come up with insecurities about the best practices for ...
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2answers
29 views

Can we apply L'hopital Rule when we have $\frac{-\infty}{+\infty}$?

I saw in a book, it evaluated $\lim_{x\to0^+}\cfrac{(\ln x)^3}{\frac1x}$ with L'hopital Rule. the fraction is $\frac{-\infty}{+\infty}$ actually. so can I use L'hopital rule safely every time I see ...
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2answers
111 views

Infinity as a limit in an indeterminate form

There is this limit $$\lim_{x\to1}\frac{\ln x}{\left(x-1\right)^{2}}$$ for which I built a graph and I know the answer, so the question is not about how to compute it, but about my observation that ...
2
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1answer
81 views

Is it logical to say that $ 2\over 0$ $\ne$ $ 2\over 0$?

As I was doing a math exercice, I came across a question which I decided to prove by contrapositive. That required me to show that $ f(4-x)$ $\ne$ $f(x)$ - but in both cases the result was $ 2\over 0$....
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What is the concept of the rate by which the limit of a function approaches a certain value?

I am having difficulty in understanding limit algebra involving infinity, and indeterminate forms. Many answers to my previous questions and those I found on the internet speak of indeterminate forms ...
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3answers
98 views

Restrictions on laws

I'm wondering about the restrictions, my doubt is for example at $\log_a(b)=c\implies a^c=b$, how would anyone add the restrictions for this? I know the argument and the base of a log have to be >0 ...
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3answers
122 views

Why is $\lim_{x \to \infty} x \log(1+\frac{1}{x}) = \lim_{y \to 0} \frac{\log(1+y)}{y}$?

Why is $$\lim_{x \to \infty} x \log(1+\frac{1}{x}) = \lim_{y \to 0} \frac{\log(1+y)}{y} ?$$ I understand that they are both indeterminate forms. Specifically we are initially given $$\lim_{x \to \...
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Shouldn't we treat the indeterminate form $\frac{0}{0}$ as $\frac{0^+}{0^+}$ while dealing with limits?

Shouldn't we treat the indeterminate form $\frac{0}{0}$ as $\frac{0^+}{0^+}$ while dealing with limits? We know that $\frac{0}{0}$ is undefined. Shouldn't it be written explicitly and clearly as the ...
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27 views

Nested Indeterminate Forms

Suppose $f$ and $g$ are differentiable functions such that $g'(x) \neq 0$ on an open interval $I$ containing $0$; $\lim_{x \to 0} f(x) = 0$ and $\lim_{x \to 0} g(x) = 0$; $\lim_{x \to 0} \frac{f'(x)}{...
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1answer
39 views

Prove identity of indiscernibles

Define the earth mover's distance as \begin{equation} EMD(\mathbf{x},\mathbf{y}) = \frac{\sum_{i=1}^m\sum_{j=1}^n f_{ij}d(x_i,y_j)}{\sum_{i=1}^m\sum_{j=1}^n f_{ij}}. \end{equation} where we define ...
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6answers
136 views

Undetermined or indeterminate forms: $\frac{0}{0}, \frac{\infty}{\infty}, 0\cdot\infty, 1^\infty, 0^0, +\infty-\infty$

I wanted to know who has decided that for the calculation of the limits of the following forms, $$\color{orange}{\frac{0}{0},\quad \frac{\infty}{\infty},\quad 0\cdot\infty,\quad 1^\infty,\quad 0^0,\...
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3answers
148 views

verifying $\lim\limits_{x \to \infty} (1+\frac{1}{\sqrt{x}})^x$

What is the limit of $\lim\limits_{x \to \infty} (1+\frac{1}{\sqrt{x}})^x$ ? I tried to solve it but I am not sure if it is appropriate to solve it this way. $(1+\frac{1}{\sqrt{x}})^x =\exp\Bigl({x\...
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2answers
119 views

In this textbook explanation of needing partial derivatives, how is this partial derivative not an indeterminate form?

$$ f(x,y) = x^\frac{1}{3}y^\frac{1}{3} $$ $$\frac{\partial f}{\partial x}(0,0) = \lim_{x \to 0} \frac{f(h,0)-f(0,0)}{h}= \lim_{x \to 0} \frac{0-0}{h} = 0$$ "and, similarly, $\frac{\partial f}{\...
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1answer
38 views

Natural log of 0, or approximate it using limits

I have a forensics question that asks when a body will reach room temperature if left undisturbed. I am told room temperature is 20 degrees and I am working with Newton's Law of Cooling. I have an ...
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4answers
86 views

Find the values of $a$ and $b$ if $\lim_{x\to -\infty}$ $\sqrt{x^2-x+1} + ax - b = 0$?

I took out x from the square root and reached the following expression, $$\lim_{x\to -\infty} x\sqrt{1-\frac{1}{x}+\frac{1}{x^2}} + ax - b = 0$$ then I separated the part of the expression which ...
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3answers
100 views

Solving indeterminate form of Limit without Using L'hopital's rule

Is there a way to solve $\lim\limits_{x \to 1} \dfrac{\ln(x)}{x-1}$ without using L'hopital?
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2answers
59 views

Evaluating $\lim_{n \to \infty} \frac{(2n)!}{2^n (n!)^2}$

Can you please explain how should I evaluate this limit $$\lim_{n \to \infty} \frac{(2n)!}{2^n (n!)^2}$$ I know the solution is $\geq1$ but I don't know how I can just simplify like this but I stuck ...
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2answers
78 views

Analysis question limit at infinity $\lim_{n \to \infty} (-1)^n \cos(n)$

1_I have questions about that when is it necessary to use bounding for limit problems e.g. this limit: $$\lim_{x\to 0} x \sin\left(\frac{1}{x}\right)$$ solution is zero because $|\sin\left(\frac{1}{x}...
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2answers
51 views

Analysis question limit $\lim_{n \to \infty} (-1)^n$

Can someone please explain to me how we should evaluate this limit, I just know that it's indeterminate form and I tried to use L'Hospital's rule but I couldn't do it here what I have done $$\lim_{n \...
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0answers
30 views

Applicability of L'Hôpital's rule

The derivation of L'Hôpital's rule requires Cauchy's theorem, which, in turn, requires the following conditions: two functions $f(x)$ and $g(x)$ must be continuous and differentiable in an interval $[...
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0answers
80 views

Why is infinity *infinity and infinity^infinity considered indeterminate?

I know that infinity is not a specific number,so we cannot apply normal algebric operations with it.But we can use the concepts of limits.So why is x^x or x*x (where x tends to infinity) indeterminate?...
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4answers
145 views

Limits to infinity of exponential functions

I am completly blocked trying to prove the solution of the limits below 1) $$ \lim _{m\to\infty}\left(\cos\left(\frac{x}{m}\right)\right)^m\\1 \quad \text{ for } a\to +\infty;\quad 0 \quad \text{ ...
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2answers
78 views

How to evaluate $\lim_{x \to \infty}\ln\left(\frac{1 - x}{1 + x}\right)$?

I need to evaluate $$\lim_{x \to \infty}\ln\left(\frac{1 - x}{1 + x}\right)$$ and I always end up with indetermined form. For example since the following holds true $$\frac{1 - x}{1 + x} = \frac{1 + x ...
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2answers
44 views

Find the value of $a+b$ if $\lim _{x\to 0}\left(\frac{\tan\left(2x\right)\:-2\sin\left(ax\right)}{x\left(1-\cos\left(2x\right)\right)}\right)=b$

Find the value of $a+b$ if $$\lim _{x\to 0}\left(\frac{\tan\left(2x\right)\:-2\sin\left(ax\right)}{x\left(1-\cos\left(2x\right)\right)}\right)=b$$ I tried plugging in $x=0$ in the function which ...
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0answers
15 views

How to deal with indeterminate values in a linear regression?

I have some experimental data over the time. I need to fit this data to two models in their linearized forms: Model 1: $\ln (a-y)=\ln a-b\ t$ Model 2: $\dfrac{t}{y}=\dfrac{1}{k\ d}+\dfrac{t}{c}$ My ...
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0answers
35 views

Could we obtain the following limit appearance?

The question looks maybe simple, but it is interesting. Let us suppose that we have the following function with the stated features: --$\,\,\,$ $F(x,a)$ was defined as below: $$ \frac{{ } \...
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1answer
94 views

Proving L' Hospital's Rule

Referring to the proof L'Hospital's first rule in Bartle and Sherbert, Introduction to Real Analysis, page 189, Fourth Edition. By the proof can one conclude that if $L$ is finite then $h(x)=L$ for ...
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2answers
185 views

Are there unsolved indeterminate limits?

I find the question itself is hard to put precisely. I apologize in advance. A simple version could be: Let $\mathcal{F}$ be the set of functions obtained via elementary binary operations (sum, ...
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4answers
82 views

Need help in evaluating an indeterminate form ($0^0$) of a limit

Let, $\Psi(x)= f(x)^{g(x)}$ Given, $\lim_\limits{x\to a} f(x)= 0, \lim_\limits{x\to a} g(x)=0$ and $\lim_\limits{x\to a} \Psi(x)$ exists. If the mentioned conditions are guaranteed to be fulfilled, ...
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1answer
13 views

point wise convergence and the indeterminate form - trivial question

I am just a beginner in Math and a little confused about the point-wise convergence. I a getting contradicting results between indeterminate form and point-wise convergence. Is it common? consider ...
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1answer
22 views

A Question About the possible values of 1^infinity

if we have $$\lim_{x\to c}f(x) = 1 $$ $$\lim_{x\to c}g(x) = \infty $$ then can $\lim_{x\to c}f(x)^{g(x)} = -\infty $ ever be true? If so, what are some examples? If not, would it be different if $ ...
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2answers
87 views

Limit of the form 0 times infinity

I'm trying to evaluate $\lim_{n\to\infty} n^3\ln\left(1+\frac{1}{n!}\right)$. It's $0\cdot\infty$ situation. I know that indeterminate forms can sometimes be evaluated using L'Hopital's rule. I ...
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1answer
66 views

Is $y=0^x$ really a function?

I know that there are three main options when it comes to dealing with $0^0$: $$0^0=0$$ $$0^0=1$$ $$\nexists x(x=0^0)$$My question is, do we have to assert that one of these conditions is true in ...
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1answer
22 views

Limit as $\alpha$ and $\beta$ tend to 1, indeterminate form

I have the equation $F=\cfrac{\beta (s-1)-(\alpha s-\beta) e^{\lambda t}}{\alpha (s-1)-(\alpha s-\beta) e^{\lambda t}}.$ such that $\lambda = \beta - \alpha$ I need to take the limit as $\alpha$ and ...
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1answer
53 views

What is the cardinality of this absolutely outrageous set?

Burning question. Consider the following set: $$A=\left\{\frac{0}{0},\frac{\infty}{\ \infty},\ 0\cdot\infty,\ 1^{\infty},\ \infty-\infty,\ 0^{0},\ \infty^{0}\right\}$$ The A stands for Any Self-...
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2answers
48 views

Using conjugates to solve indetermination on limit with cubic root

I have the following problem: $$\lim\limits_{x \to -8} \frac{\sqrt{1-x} -3}{2 + \sqrt[3]{x}}$$ which seems at first like a simple limit with square and cubic roots that I can solve using conjugates ...
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3answers
113 views

Continuity of $x\ln(x)$

Why does the function $f(x)=x\ln(x)$ is well-behaved at $x=0$? Should zero not belong to the domain of $f$, since we have natural log function? I tried to compute the limit of $f$ when it goes to ...
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3answers
135 views

If $0^0 = 1$, then is it true that $0/0 = 1$?

By Knuth, Concrete Mathematics (2nd ed.) page 162, it is convenient that $$0^0 = 1$$ Then, is it true that $$0^1/0^1 = 0^{1-1}= 0^0 = 1$$ and we are free to exclude indeterminate statement of $0/0$...
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4answers
132 views

An indeterminate limit form of infinity/infinity

I am trying to solve the limit: $$\lim_{x\to\infty}x^\frac{5}{3}\left(\left(x+\sin\left(\frac{1}{x}\right)\right)^\frac{1}{3}-x^\frac{1}{3}\right)$$ I was trying to find a way to bring it into a ...
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3answers
66 views

Why is $\infty^0$ an indeterminate form?

I can't seem to find a situation where $\lim_{x\to a}f(x)=\infty$ and $\lim_{x\to a}g(x)=0$, but $\lim_{x\to a} f(x)^{g(x)}\neq 1$. This makes me think that, even though this form ($\infty^0$) is ...
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2answers
92 views

Evaluating $\lim\limits_{x\to 1}\frac{2^{\sin(\pi x)}-2^{\tan(\pi x)}}{x-1}$

I am stuck at evaluating this limit: $$\lim\limits_{x \to 1}\frac{2^{\sin(\pi x)}-2^{\tan(\pi x)}}{x-1}$$ Can someone help me, please? Also, I am not supposed to use L'Hospital's rule or derivatives. ...
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1answer
66 views

$\sqrt[0]{x}$ indeterminate form?

I stumbled across a list of indeterminate forms on Wolfram Alpha. On top of the common ones, plus a few ones like $(-1)^\infty$ that usually aren't mentioned in textbooks but I can make sense of, one ...
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4answers
94 views

How can I turn a $0^0$ form into a $\frac{\infty}{\infty}$ or $\frac{0}{0}$ form?

I am trying to evaluate this limit: $$\lim_{x\to0^{+}}(x-\sin x)^{\frac{1}{\log x}}$$ It's a $0^0$ intedeterminate form, and I am unsure how to deal with it. I have a feeling that if I could turn it ...

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