# Questions tagged [limits]

Questions on the evaluation and properties of limits in the sense of analysis and related fields. For limits in the sense of category theory, use the tag “limits-colimits” instead.

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### Trying to prove $(a^x)^y=a^{xy}$ using definition

In a lecture, we learned that $a^x$ is defined to be the limit of $a^{x_n}$ where $x_n$ is a series of rationals that converge to x. I was trying to prove $(a^x)^y=a^{xy}$ using solely this definition,...
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### Showing that the function that returns $1$ on the rationals and $0$ elsewhere has no limit at $0$. Is there anything wrong in this proof? [duplicate]

Define $f : \mathbb R \to \mathbb R$ by $f(x) = \begin{cases}1, & \text{if } x\in \mathbb Q \\ 0, & \text{if } x \notin \mathbb Q \end{cases}$. I want to show that this function has no limit ...
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### Evaluating $\lim_{x\to\infty}\left[\cos\frac1x\right]^{h(x)}$, where $h(x)=\frac{x^4+x^2-1}{2x+1}\sin\frac1x$

I am struggling to calculate what $h(x)$ tends to in $$\lim_{x\to\infty}\left[\cos\left(\dfrac{1}{x}\right)\right]^{h(x)}$$ where $$h(x)=\dfrac{x^4+x^2-1}{2x+1}\sin\left(\dfrac{1}{x}\right)$$ This is ...
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### Why can the Binomial Distribution be Approximated by a Normal Distribtuion?

As a practice problem, I am trying to prove the relationship between the Normal Distribution and the Binomial Distribution. I have seen several proofs of this before (e.g. Justifying the Normal Approx ...
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### Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.

The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as $$\log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2}$$ where $\arg(z)$ is the standard branch of the ...
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### Vector algebra and limits question

The following question just occurred to me yesterday: The zero or null vector, $\boldsymbol{\vec 0}$, has neither magnitude nor direction, but for a unit vector $\boldsymbol{\vec r}$ and some scalar ...
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### Is $\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$ not convergent?

I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$$ where $\{x\}$ denotes the fractional part of $x$, $n!$ is the factorial of $n$...
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### How to solve this limit $\lim_{n\to \infty } \sum_{k=0}^n 1/[(2k+1)(2k+3)]$ [closed]

Help please: $$\mbox{I tried to do it like this}\quad \lim_{n \to \infty}\sum_{k = 0}^{n}{1 \over \left(2k + 1\right)\left(2k + 3\right)}$$ but I don't know how to continue.
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### Solution verification of $\lim_{n \to \infty} {x}^{\left( {n}^{x / n} \right)}=x$. [closed]

Let $x\in\mathbb{R^{+}}$ and $n\in\mathbb{N}$. Consider the function $$f(x)=x^{\left(n^{x / n} \right) }.$$ For $x=0$, $$\lim_{n\to\infty} f(x) =\lim_{n\to\infty} 0^{(n^{0})}=0.$$ For $x≠0$ and ...
1 vote
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### Evaluating a one-sided limit that goes to negative infinity, where the denominator goes to $0$

I attached my handwriting explaining the issue, hope that's okay. Basically, I can "intuitively" find the limit and explain it on paper. However, I'm not sure if this "approach" is ...
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### Finding a weaker speed of divergance for a real sequance [closed]

Let \begin{equation*} \lim_{n} \frac{1}{n} \sum_{k=1}^{n-1} \frac{\ell_{k-1}}{\ell_{k}} \leq \gamma <1,~ and ~~ \frac{\ell_{k-1}}{\ell_{k}} \leq \frac{3}{2} ~ \forall ~ k \in \mathbb{N} \end{...
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Short Version: I am interested in computing (as a closed form) the limit if it does exist: $$\lim_{k \rightarrow \infty} \left[\sum_{a^2+b^2 \le k^2; (a,b) \ne 0} \frac{1}{a^2+b^2} - 2\pi\ln(k) \... • 17.5k 7 votes 3 answers 418 views ### Showing that f(x)=\lim_{n\to{\infty}}(\frac{x}{n}+1)^n has the property f(a+b)=f(a)\cdot f(b) On the way to show that the function f(x)=\lim_{n\to{\infty}}\left(\frac{x}{n}+1\right)^n has this property: f(a+b)=f(a)\cdot f(b), a math professor explained me that: \lim_{n\to{\infty}}\left(\... • 277 0 votes 2 answers 198 views ### Proving that a function with only removable discontinuities can be made continuous I'm working with Spivak's "Calculus" and was doing the following problem: Let f be a function with the property that every discontinuity is a removable discontinuity. This means that \... • 281 2 votes 4 answers 111 views ### Is it possible to show that for q>0, \lim\limits_{x\to\infty}\dfrac{(\ln{x)^p}}{x^q} = 0 without using L'Hopital's Rule? Is it possible to show that for q>0, \lim\limits_{x\to\infty}\dfrac{(\ln{x)^p}}{x^q} = 0 without using L'Hopital's Rule? Applying L'Hopital's Rule repeatedly until the numerator becomes a ... • 1,426 0 votes 0 answers 115 views ### Prove that this limit is equal to \sqrt{2} for the function f(x)=x^2-2 for an arbitrary seed point s. Mathematica knows that:$$ s + \frac{1}{1-\lim_\limits{n\ \to\ \infty}\left[\frac{\displaystyle\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{f\left(k/n + s -1/n\right)}}{\displaystyle\sum _{k=1}^...
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Hey i have this function, and I don't understand why I get wrong limit if I insert x into the square root, even though it's correct algebraic to insert it. $$\frac{\sqrt{x^2 + 9}}{x}$$ The first ...
### $\lim_{n \rightarrow \infty} \frac{1}{n^2} \sum_{k=1}^{n} \frac{k}{\ln(k+1)}$ [duplicate]
I am interested in the limit $\lim_{n \rightarrow \infty} \frac{1}{n^2} \sum_{k=1}^{n} \frac{k}{\ln(k+1)}$. While I suspect the limit to be 0, I cannot prove it rigorously. By intuition we have \$\sum_{...