# 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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### Can we say that $1^\infty$ is indeterminate because we can't classify $\infty$ as even or odd?

While studying indeterminate forms I like many others had used this post to understand why $1^\infty$ is indeterminate. Today I thought of a new and perhaps simpler argument to explain why this is so ...
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### How to get the limit of the sequence $(\sqrt{n^2+n}-\sqrt{n^2-1})$? [duplicate]

My Attempt $$\sqrt{n^2+n} - \sqrt{n^2-1} = \sqrt{n+1} \, \bigl(\sqrt{n}-\sqrt{n-1} \bigr)$$ Then I tried to apply the sandwich theorem in some way but failed. Important Note Please do not solve the ...
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### Using L'Hospital's Rule for Fourier's series

The origin of this question comes from watching this YouTube video at 8:00. The equation to evaluate is $$f(x)=\frac{1}{i(j-k)}e^{i(j-k)x}|_{-\pi}^{\pi}$$ However, this equation can be evaluated ...
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### Does $\lim_{x \to 0^+} (1 + x)^{\ln x}$ exist? [closed]

Does $\displaystyle \lim_{x \to 0^+} (1 + x)^{\ln x}$ exist? I do not have any attempts solving it because I do not know how to transform it into the form in which I use l'Hôpital's rule.
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### Can the following $\infty - \infty$ limit be solved without using l'Hôpital rule?

I know that some of the $\infty - \infty$ limits can be solved without using l'Hopital's rule. In such cases, you would usually be able to either rationalize (or derationalize if that is what its ...
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### Bypassing the indeterminate form $0\cdot \infty$

Trying to calculate out the limit \begin{array}{rcl} \lim_{x \to +\infty } (e^{x^{2}\sin \frac{1}{x}}-e^{x})&& \\ \end{array} I come up with the indeterminate form $0\cdot \infty$ as ...
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### How can we multiply&divide the expression by its conjugate in the $\infty-\infty$ indeterminate case? Is it not $\frac{\infty}{\infty}$?

Question $$\lim _{x \rightarrow \infty} \left(x-\sqrt{x^2+5 x}\right)$$ To evaluate the limit, we multiply and divide the expression by its conjugate. First Question But since $x \rightarrow \infty$, ...
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### $\infty-\infty$ indeterminate case

The methods yield two different answers. Could you explain the reason clearly and in detailed? Question: $$\lim_{{x \to \infty}} \left( \sqrt{x^2 + 6x + 14} - (x+1) \right) = ?$$ Solution: Method 1: ...
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### Showing that this piecewise defined function is infinitely differentiable [duplicate]

Let $a<b$ and $f(x) = 0$ if $x$ is outside $(a,b)$ and $f(x) =e^{-\frac{1}{x-a}}e^{-\frac{1}{b-x}}$ is $a<x<b$. Show that $f$ is infinitely differentiable on $\mathbb R$ If $x\neq a,b$ then ...
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### Indeterminacy of 1^(infinity)

I am being taught that 1^(infinity) is indeterminate. But I couldn't stop thinking why 1x1x1x1....=1 is incorrect. Please tell me if my reasoning is correct? I thought like this. So, for limit to ...
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### How can I rewrite a function so it’s no longer in indeterminate form when $x$ is in the exponent?

I have to rewrite $f(x)$ so that it is no longer is indeterminante as the limit approaches infinity. $$f(x) = \frac{6+2e^{-5x}}{5-16e^{-5x}}$$ I am really not sure how to go about it, though. Can ...
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### Difficult limit question involving Euler's number and L'Hospital

The limit I'm trying to evaluate is $$\lim_{x\to+\infty} e^x \left[e - \left(1 + \frac{1}{x}\right)^x\right]$$ After some hours trying, I've made almost no progress. I always end up in some ...
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### Does $\frac{\sin(tx)}{\sin(x)}$ have a name?

Does the following function have a name? $$\operatorname{boxySine}(t,x) = \begin{cases} \frac{\sin(tx)}{\sin(x)} & x \neq 0 \\ t & \text{otherwise} \end{cases}$$ It appears in ...
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### Doubt regarding L' Hospital rule

I had calculated $\lim_{z\to 1} (1-z).\tan(πz/2)$ by L'Hospital Rule as Follows I took $\tan(πz/2)$ to the denominator of the denominator i.e $(1-z)/1/\tan(πz/2)$ which becomes $(1-z)/\cot(πz/2)$ and ...
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### How is Taylor's Theorem Related to Maxima Minima and Indeterminate Forms ( Relationship between them )

I'm reading a real Analysis book in which Taylor's Theorem was introduced in chapter 10 and now the next two chapters are 11. Maxima and Mamina and 12. Indeterminate Forms. In both the chapters the ...
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### Applying theorems and properties to solve indeterminate limits

I have been trying to solve the following limit for my university Mathematical Analysis class: $$\lim \limits_{x \to +\infty}\sin(x)(\ln{(\sqrt{x}+1})-\ln(\sqrt{x+1}))$$ I know that the answer is 0, ...
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### Useful theorems/properties for evaluating $\lim \limits_{x \to +\infty}\sqrt{x}(\sqrt{x+1}-\sqrt{x-1})$

I have been trying to solve the following question from my Mathematical Analysis course in university: $$\lim \limits_{x \to +\infty}\sqrt{x}(\sqrt{x+1}-\sqrt{x-1})$$ I am aware that the answer is 1, ...
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Is it true that $$\displaystyle\lim_{x\to 0}x\int_{0}^\infty k^5 dk=0$$ Seems like this has an indeterminate form, but I am being told it equals zero.
I did the following problem: Find $\displaystyle\lim_{x\to 0} \frac{\cos^2x-1}{x^2}$ The following solution was given: \lim_{x\to 0} \frac{\cos^2x-1}{x^2} = \lim_{x\to 0} \frac{2 \sin x \cdot \cos x}...