# Tagged Questions

Questions on the evaluation of limits.

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### How to get out of this indetermination $\lim\limits_{x\to -1} \frac{\ln(2+x)}{x+1}$?

Well, is just that, I can't remember a way to get out of this indetermination (with logarithm), can someone help me? I'm studying for my calculus test and this question is taking me some time. ...
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### How to find the limit without L'Hospital?

$$\lim_{x\rightarrow 1} \frac{x^m-1}{x^n-1}$$ I found this in a book (recommended by the professor) without solutions or hints. My problem is that I can't find definitions about this kind of ...
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### limit $a_{n+1}/a_n$ for recurrence $a_{n+2}=a_{n+1}+a_n$

Let $\{a_n\}$ be a positive sequence which satisfies $a_{n+2}=a_{n+1}+a_n$ for $n=1,2,\ldots$. Let $z_n=a_{n+1}/a_n$. How can I prove that $\lim_{n\rightarrow\infty}z_n$ exists? I looked at ...
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### Showing that $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$

I need prove the following: ($\zeta(s)$ is the Riemann zeta function) $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$ I really don't know, i have tried, but nothing for now.
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### What is the answer to the following limit?

What is the answer to the following limit? $$\lim_{x\rightarrow +\infty} \sum_{k=0}^\infty \frac{(-1)^k x^k}{k!} \frac{(4N+2k-1)!(k+N-1)!}{k!(k+2N-1)!^2}$$ where $N> 1$.
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### How i can determine: $\lim_{x\to1} \frac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$?

This is actually a limit tending to 1, if you can help me see how are the steps to multiply the factors, because it seems that there are many multiplications and this confuses me a lot! ...
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### Need to show that $\lim_{x\to\infty}\left(\sum_{n\le x}^{}\frac{1}{n}-\ln x \right)$ exist and is less than $1$ [duplicate]

Need some help here. I need prove that the following limit exist and is less than $1$ $$\lim_{x\to\infty}\left(\sum_{n\le x}^{}\frac{1}{n}-\ln x\right)$$ I feel a little lost here, this is my first ...
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### Find $\lim_{x\to-\infty}{x+e^{-x}}$

I have this exercise in my worksheet: $$\lim_{x\to-\infty}{x+e^{-x}}$$ I am always ending up with $-∞+∞$ or $\frac{∞}{∞}$. It says the answer is $+∞$, but how can I get that?
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### Can I use l'Hôpital to find a limit after changing to polar coordinates?

I was wondering if there is another approach instead of using the $log (x + 1) \sim_{0} x$ or Taylor series to solve: $$\lim_{(x,y) \to (0,0)} f(x,y)={\ln(xy + 1) \over x}.$$ Particularly, I'm ...
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### $\lim_{n\to\infty} a_n=a$ if and only if $\forall p\in \Bbb N$, $\lim_{n\to\infty} |a_{n+p}-a_n|=0$

I'm doing exercises. In the related book, there is a claim. Is this right? I'm not sure. For a sequence $\{a_n\}$, there exists a limit $a$ such that $\lim_{n\to\infty} a_n=a$ if and only if for ...
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### For what values ​​of $m$ the function $y=x^m\sin(x)$ have horizontal asymptote

I want to figure for what values ​​of $m$ the function have horizontal asymptote.$$y=x^m\sin(x)$$ so what I understand from that is this that the function dont have a vertical one, so I will find ...
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### Calculate the limit of $\lim\limits_{x\to 1^-}\left(\frac{1}{1-x^2} -\frac{1}{1-x^3}\right)$

I want to calculate this limit and wonder what is the best way to calculate it. $$\lim\limits_{x\to 1^-}\left(\frac{1}{1-x^2} -\frac{1}{1-x^3}\right)$$ I tried to do the following thing ...
For the function $\displaystyle h \sin \left(\frac{1}{h}\right)$ when it is evaluated at $h=0$, is it $0$ or is it undefined?