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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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Calculating indeterminate form limits involving $\cos(x)$ and $\sin(x)$, using only algebraic manipulation

I was doing some calculus homework and I came across with some problems. I have to find the following limits 1) $\displaystyle\lim_{x\to \frac{\pi}{2}} \frac{\sin(x)-1}{\cos(x)}$ 2) $\displaystyle\...
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2answers
55 views

why 1/ infinity isn't indeterminate like other indeterminate?

$1/\infty$ tends to 0. $\mathbf {It\ doesn't \ satisfy\ the\ inverse \ process\ of\ multiplication \ and \\division\ i.e} $ $\infty * 0$ is undefined or indeterminate. So why $1/\infty$ is not ...
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1answer
44 views

Prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$ [duplicate]

I have no idea how to proceed with proving this. If $X$ is a continuous random variable, $P(X > 0) = 1$, $E(X)$ is defined and $F(x)$ is the CDF, then prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$
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4answers
71 views

L'hopitals rule gone wrong? [closed]

So I was thinking that if you do $ \lim_{x\to \infty} 0x$ the answer is obviously $0$ right? The numbers getting bigger don't matter because the multiplication by $0$ just turns them into $0$. But you ...
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1answer
54 views

False Proof of $0^0=1$ [duplicate]

I saw the following argument on Quora that provides which should actually be a false proof since $0^0$ is not defined to be $1$. $(a+b)^{n}=\sum_{k=0}^{n}\binom n{ k}a^kb^{n-k}$. Put $a=0, b=1, n=...
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0answers
47 views

Matrix geometric sum with a unit eigenvalue

Let $A$ be a complex, square matrix, and define the geometric sum $$S = I+A+\cdots + A^{N-1}. \tag{1} $$ Just like in the scalar case, one can expand and see that $$(A-I)S =A^N-I, \tag{2} $$ and hence,...
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3answers
70 views

infinity mathematics

what is the result of (approaching infinity)/(approaching zero) ? I think its approaching infinity, but if it is approaching infinity, then multiplying both sides by approaching zero, it became: (...
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47 views

Laplace Transform: Indeterminate Form in Definite Integral Change of Variables Calculation

I was trying to find the Laplace transform of $e^{3t}$: $$\int^\infty_0 e^{3t}e^{-st} \ dt = \int_0^\infty e^{3t - st} \ dt = \lim_{x \to \infty}\int_0^x e^{3t - st} \ dt$$ So if we then attempt to ...
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4answers
60 views

$\lim\limits_{n \rightarrow +\infty} \frac{\ln(1+n+n^3)-3\ln(n)}{n(1-\cos(1/n^2))}$

I want to solve this limit: $$\lim_{n \rightarrow +\infty} \frac{\ln(1+n+n^3)-3\ln(n)}{n(1-\cos(1/n^2))}$$ I have proved that $\lim\limits_{n \rightarrow +\infty} \frac{\ln(1+n+n^3)-3\ln(n)}{n} = 0$ ...
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Is the meaning of *indeterminate* in the context of polynomial theory the same as in the context of, say, L'Hopital's rule?

This question is a follow-on to Is "indeterminate" a synonym for "variable" or for "transcendent"? . I have reproduced the quotations and refined some of my original ...
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3answers
44 views

Limits with Taylor series around zero

I had some problems with the following two limits, which are supposed to be calculated with Taylor series: $$ \lim_{x\to 0^+}\frac{e^\sqrt{x}-e^{-\sqrt{x}}}{\sqrt{\sin{2x}}}\quad\mbox{and}\quad \lim_{...
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3answers
49 views

Are $\log_1 1$ and $\log_0 0$ indeterminate forms?

Are $\log_1 1$ and $\log_0 0$ indeterminate forms? Whenever I ask someone about these indeterminate forms, they deny by saying either $\log$ is neither defined at base $0$ nor at base $1$, or they ...
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2answers
87 views

Limit and L'hopital's Rule

How do you evaluate the following limit? $$\lim_{x \to 0} \dfrac{x\sin^{-1}x}{x-\sin{x}}$$ When I is L'Hopital's rule twice, I get: $$\lim_{x \to 0} \dfrac{(x^2+2)\csc x}{(1-x^2)^{3/2}}$$ Which doesn'...
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4answers
74 views

Limlit of $\frac{x-x\ln(1+x)-\ln(1+x)}{x^3+x^2}$ at $0$ without l'Hôpital or Taylor series

I have $f : x \mapsto \frac{\ln(1+x)}{x}$ which derivative is $$\frac{1}{x(1+x)}-\frac{\ln(1+x)}{x^2}$$ I want to find the limit as $x$ goes to $0$ of this derivative. I've tried simplifying the ...
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1answer
52 views

Notation Question for “Generalization of L’hopital’s Rule” PDF by V. V. Ivlev and I. A. Shilin

Within this brief text starting on page 2, the author(s) have a habit of employing a sort of "mixed" notation of differentiation, mixing the "prime marks" of Lagrange's notation for differentiation of ...
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1answer
37 views

Indeterminate form of limit: is it obligatory to write it?

I'm preparing for a Limits/Derivatives exam. I have two very basic questions that I couldn't find an answer to. Question 1: when calculating limit of a sequence or of a function, do I always have to ...
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1answer
84 views

Improper integral over the rationals

Question: Suppose that I wish to integrate a function over the natural numbers. How could I do this? Answer: Consider the definite integral $\int_a^bf(x)\ dx$. If we consider this as the 'area ...
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2answers
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What is the limit of the following expression?

I've been thinking about and trying to solve the following limit that I just feel lost by now. I always get an indeterminate form. I don't know what else to try. In the picture is just one way that I ...
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6answers
70 views

Limits - Calculating $\lim\limits_{x\to 1} \frac{x^a -1}{x-1}$, where $a \gt 0$, without using L'Hospital's rule

Calculate $\displaystyle\lim\limits_{x\to 1} \frac{x^a -1}{x-1}$, where $a \gt 0$, without using L'Hospital's rule. I'm messing around with this limit. I've tried using substitution for $x^a -1$, ...
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8answers
88 views

$\infty\cdot 0$ Indetermination without L'Hopital?

Evaluate $\lim_{n\to\infty}n\cdot r^n$ , being $0<r<1$. I dont know if I took the proper steps, but I get to this point: $$\lim_{n\to\infty}n\cdot r^n=\lim_{n\to\infty}n \lim_{n\to\infty}r^n = ...
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2answers
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Limit of sequence is indeterminate.

We have to find the limit of the sequence $a_k = \left(\dfrac{k^4 11^k + k^9 9^k}{7^{2k} +1}\right)$. Here is my attempt: $$\left(\frac{k^4 11^k + k^9 9^k}{7^{2k} +1}\right) = \frac{k^4 11^k + k^9 9^...
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1answer
73 views

System of equations for which Cramer's rule is invalid

My teacher told me, that there exists a system of $3$ equations with $3$ unknowns, which isn't indeterminate despite all determinants being equal to $0$. Is it true? How to find this system?
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1answer
32 views

How many terms are 'missing'?

I know in this particular indeterminate partial sum S = $3^n - 3^{n+1} + 3^{n+2} - 3^{n+3} + \cdots + 3^{3n}$ where $a=3^n$ and $r=-3$. So I know if $3^1$ were the first term, there would be $3n$ ...
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2answers
66 views

If f(x,mx) is algebraically reduced to contain only 'm', does it prove the limit does not exist?

Suppose you trace the path y=mx of a multivariable function f(x,y) to find the limit as (x,y)->(0,0). If f(x,mx) is algebraically reduced to contain only 'm', does it prove the limit does not exist? ...
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1answer
92 views

$\sqrt{3} - 0m = 0$ only if $m = \infty$ … What?!

On "Introduction to Discrete Dynamical Systems and Chaos" by Mario Martelli, at 86 pp, it's stated the implication used as title for this question: $$"\ldots\sqrt{3} - 0m = 0 \text{ , which can be ...
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2answers
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Limit result is wrong compared with specific values asigned

I have two expressions: $t_1$ and $t_2$. I want to calculate division $t_1 / t_2$ when the parameters $r_1 = r_2$. This condition $r_1 = r_2$ causes the denominators of $t_1$ and $t_2$ to be ...
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2answers
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Limits, derivatives, and dividing by zero. [Contradiction in derivative defintions?]

In the limit definition where the denominator is $x - a$, and we take the limit as $x$ approaches $a$, we assume that this denominator is not equal to zero. Where (besides the fact that it is ...
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3answers
47 views

Evaluating $\lim_{x\rightarrow\infty}\left(\frac{2x-1}{3x+2}\right)^x$.

I have been trying to solve this limit but i think it doesnt get me anywhere. I tried with ln(y) but nothing. I tried to transform it to inf/inf but no result . Can anyone please help me find it ...
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2answers
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Evaluate: $\lim_{x \to \pi/4}\frac{8-\sqrt{2}(\sin x+\cos x)^5}{1-\sin 2x}$ without L'Hopitals

I have to find the limit of $\lim\limits_{x \to \pi/4}\frac{8-\sqrt{2}(\sin x+\cos x)^5}{1-\sin 2x}$ here is my try $\lim\limits_{x \to \pi/4}\frac{8-\sqrt{2}(\sin x+\cos x)^5}{1-\sin 2x}=\lim\...
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1answer
738 views

Indeterminate form of infinity over 0?

I know that indeterminate forms exist in limits, such as $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0^0$, $\infty^0$, $1^\infty$. Then, if $\lim\limits_{x \to a} p(x)=\infty$ and $\lim\limits_{x \to a}...
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4answers
104 views

Is $0^i$ indeterminate? [closed]

According to Euler's identity $$e^{ix}=\cos{x}+i\sin{x}$$ Using this identity $$0^i=e^{i\ln{0}}=\cos\ln0+i\sin\ln0=\cos{-\infty}+i\sin{-\infty},\because \ln0=-\infty$$ But $\cos{-\...
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4answers
45 views

Limits , indeterminate form

$$(\lim_{x\to 0} (\cos2x)^{\frac 2{x^2}} = e^{-6})$$ My doubt is that why it's indeterminate? We can see $x \neq 0$, i.e., it is tending to $0$. So $\cos2x$ cannot give a value equal to $1$ or greater ...
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3answers
65 views

Why a number which is in (-1,0) raised to infinity is 0?

I hope it's not a duplicate but I've been searching about this problem for some time on this site and I couldn't find anything. My problem is why a number $\in(-1,0)$ raised to $\infty$ is $0$. For ...
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1answer
189 views

is $\frac{0}{0}$ indeterminate or undefined? [duplicate]

I know in calculus the form $\frac{0}{0}$ is indeterminate.but if it is not calculus is it still indeterminate or undefined in the real number field? P.S. I know that $\frac{1}{0}$ is undefined ...
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0answers
36 views

Matrix Square Root Singularity

I have a smooth function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ with $\nabla f = [f_x, f_y]$ and a matrix $$ M = (1 + f_x^2 + f_y^2)^{-1} \begin{bmatrix} 1 + f_y^2 & -f_xf_y & 0 \\ -f_xf_y ...
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1answer
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Calculate the limit $\lim_{x\to0^+}\frac{\sqrt{\cos(2x)}-\sqrt{1+x\sin(x)}}{\tan^2\frac x2}$ [closed]

$$\lim_{x\to0^+}\frac{\sqrt{\cos(2x)}-\sqrt{1+x\sin(x)}}{\tan^2\frac x2}$$ I need to calculate this limit but I don't know what to do for to get out this indeterminate. I make: $\sqrt{\cos(2x)}-\...
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3answers
77 views

How do I prove $\displaystyle\lim_{x\to -∞} (x\cdot e^x)=0$ using L'Hôpital's Rule?

The above limit can be written as: $\displaystyle\lim_{x\to -∞} (x\cdot e^x)=\displaystyle\lim_{x\to -∞} \frac{e^x}{1/x} $. The limit is an Indeterminate type of ${0/0}$. It can be solved using L'...
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2answers
106 views

Why $\lim_{n\to \infty} \left(1+ \frac 1n \right)^n=e$ doesn't imply $\lim_{n\to \infty} \left(1+\frac 1n \right)≠1?$ [duplicate]

I'm looking for the answer to this question. But I could not find the "satisfactory" answer. This is obvious, $$\lim_{n\to \infty} \left(1+\frac 1n \right)=1+0=1$$ and $$\lim_{n\to \infty} \...
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2answers
87 views

Potential and factorial limit that tends to infinity

I've a question about this limit. As you can see it has a potential argument and a factorial one as well. $$\frac{n^p}{n!} $$ ($p$ belongs to $N_1$) When the limit tends to infinity, the ...
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2answers
74 views

Is there a way to prove or define or delineate all the indeterminate forms?

Is there a way to know what any given indeterminate form looks like? Can we prove there are only so many finite cases of them? On Wikipedia I see $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$, $0 \...
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0answers
71 views

What are all the indeterminate forms and what are some well known examples of showing their indeterminacy?

I'm assuming this is an exhaustive list of indeterminate forms: $$\infty -\infty, \frac{0}{0}, \frac{\infty}{\infty}, 0 \cdot \infty, 1^\infty, \infty^0, 0^0$$ Are there canonical examples that show ...
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6answers
47 views

How to solve $\lim_\limits{x\to1} \frac{\sqrt{2x-1} -1}{x^2-1}$?

If I substitute 1 to all the $x$ I get $\frac{0}{0}$. So I thought to factorize the expression. I can factorize the denominator $x^2-1$ and it becomes $(x+1)(x-1)$ but I don't know what to do with the ...
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0answers
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Indeterminate forms: Usual and unusual [duplicate]

What are indeterminate forms? And how are some usual and unusual? I know that indeterminate forms can be the ratio of two functions where the functions have a zero limit tendency, but I do not fully ...
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4answers
79 views

Computing : $\lim_{x\to \infty} \frac{\sqrt{4x^2+5}-3}{\sqrt[3]{x^4}-1} $ [closed]

Can you please help me with this limit? I can´t use L'Hopital rule. $$\lim_{x\to \infty} \frac{\sqrt{4x^2+5}-3}{\sqrt[3]{x^4}-1} $$
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1answer
131 views

Can one apply LHopitals' rule to differentiable functions defined over the naturals?

For e.g, if we have $\lim_{n \rightarrow \infty} \frac {f(n)}{g(n)}$= $\frac 00$, $f:\mathbb {R} \rightarrow \mathbb {R}$ and $g:\mathbb {R} \rightarrow \mathbb {R}$ (note that $f$,$g$ are defined on $...
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1answer
99 views

Limit of incomplete Beta function $\lim_{x\to \infty} \frac{B_\alpha(a+x,b)}{\alpha^x}$

I wish to prove that the following limit goes to zero: $$\lim_{x\to \infty}\frac{B_\alpha(a+x,b)}{\alpha^x} \to 0, \qquad \forall 0 \leq \alpha \leq 1$$ where $B_\alpha(a+x,b) = \int_0^\alpha t^{a-1}(...
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1answer
761 views

What is undefined times zero?

Einstein's energy equation (after substituting the equation of relativistic momentum) takes this form: $$E = \frac{1}{{\sqrt {1 - {v^2}/{c^2}} }}{m_0}{c^2} % $$ Now if you apply this form to a ...
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8answers
2k views

Find the limit of $\lim_{x\to0}{\frac{\ln(1+e^x)-\ln2}{x}}$ without L'Hospital's rule

I have to find: $$\lim_{x\to0}{\frac{\ln(1+e^x)-\ln2}{x}}$$ and I want to calculate it without using L'Hospital's rule. With L'Hospital's I know that it gives $1/2$. Any ideas?
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5answers
91 views

Calculate the limit: $\lim_{x\to+\infty}(\frac{x^2 -x +1}{x^2})^{\frac{-3x^3}{2x^2-1}}$ without de l'Hôpital rule

I was wondering how can I calculate the limit: $$\lim_{x\to+\infty}\left(\frac{x^2 -x +1}{x^2}\right)^{\frac{-3x^3}{2x^2-1}}$$ without de l'Hôpital rule. I tried to reconduct the limit at the well ...
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3answers
91 views

Calculate limit: $\lim_{x\to0} \frac{\log(1+\frac{x^4}{3})}{\sin^6(x)}$ without de l'Hôpital rule

I want to calculate the limit: $$\lim_{x\to0} \frac{\log(1+\frac{x^4}{3})}{\sin^6(x)} $$ Obviously the Indeterminate Form is $\frac{0}{0}$. I've tried to calculate it writing: $$\sin^6(x)$$ as ...