# 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.

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

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 $\lim\limits_{n \rightarrow +\infty} \frac{\ln(1+n+n^3)-3\ln(n)}{n} = 0$ ...
1answer
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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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### 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 ...
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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### 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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### 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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### 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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### 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 ...
4answers
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### Computing : $\lim_{x\to \infty} \frac{\sqrt{4x^2+5}-3}{\sqrt{x^4}-1}$ [closed]

Can you please help me with this limit? I can´t use L'Hopital rule. $$\lim_{x\to \infty} \frac{\sqrt{4x^2+5}-3}{\sqrt{x^4}-1}$$
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### 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 ...
8answers
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### 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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### 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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### 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 ...