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 (limits-colimits) instead.

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Conjecture regarding sequence of improper integrals of form $\int_1^\infty \frac{\ln x}{f(x)}dx$

I was playing around with integrals of the form $$\int_1^{\infty}\frac{\ln x}{f(x)}dx$$ and noticed something interesting when $f(x)=x(x-1)$. By numerical computation the answer tends to $1.644\ldots$ ...
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Help evaluating $\lim_{x \to 0}{\frac{a^{k} - a^{k - x}}{bx}}$ without L'Hospital?

Evaluate $\displaystyle\lim_{x \to 0}{\frac{a^{k} - a^{k - x}}{bx}}$ if $a$, $b$, and $k$ are positive real numbers. For this attempt, let $L$ be the limit, if it exists. Then, \begin{align*}L &\;...
Let $N\geq 1$ and let $F:\mathbb{R}^N\to \mathbb{R}$ be a continuous, radially symmetric and nonnegative function such that: (i): a\in\mathbb{R} \mbox{ exists such that } \liminf_{r\to +\infty}\frac{...
Does anyone know how to prove this problem. If $\{a_n\}$ is a convergent numerical sequence, and $b$ is a real number that appears in infinit number in $\{a_n\}$, then limit of $\{a_n\}$ when $n$ goes ...