query: Limit of the nth root of integral of $f$ is $\sup f(x)$
positive_passages.docid: 151568
positive_passages.text: Limit Integral equal to a supremun: $\lim_{n\to \infty} \left[\int_a^b (f(x))^n dx\right]^{{1}/{n}}=\sup_{x\in [a,b]} f(x).$
negative_passages.docid: 119145
negative_passages.text: $\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$
negative_passages.docid: 82567
negative_passages.text: Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $
negative_passages.docid: 44054
negative_passages.text: Evaluating the integral $\mathop{\lim}\limits_{n \to \infty} \int_{-1}^{1} f(t)\cos^{2}(nt) \ dt $
negative_passages.docid: 66824
negative_passages.text: Two definitions of $\limsup$
negative_passages.docid: 150896
negative_passages.text: Show that $\lim_{p \to \infty} \| f \|_p = \| f \|_\infty$
negative_passages.docid: 97256
negative_passages.text: For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$
negative_passages.docid: 35325
negative_passages.text: Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
negative_passages.docid: 45806
negative_passages.text: Finding $\lim\limits_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$ for continuous $f:[0,1]\to[0,\infty)$
negative_passages.docid: 96848
negative_passages.text: Prove: $\limsup_{n\to\infty} (a_n)\leq \sup(a_n)_{n=1}^\infty$
negative_passages.docid: 13732
negative_passages.text: $||f||_p\to ||f||_\infty$ under general assumptions
negative_passages.docid: 113141
negative_passages.text: Prove that $\lim_{n \to \infty}\bigg[\int_0^1f(t)^n \text{dt}\bigg]^{1/n}=M$
negative_passages.docid: 32742
negative_passages.text: The limit of $\frac{|A_n|}{n^2}$
negative_passages.docid: 117640
negative_passages.text: Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $
negative_passages.docid: 31931
negative_passages.text: limit as $x$ approaches infinity of $\frac{1}{x}$
negative_passages.docid: 40562
negative_passages.text: For $f :\mathbb{R} \rightarrow \mathbb{R}\displaystyle$ we have ${\lim_{n \to +\infty} f(x+n) =0}$ almost everywhere
negative_passages.docid: 157485
negative_passages.text: Integral of limit of pointwise convergent sequence of non-negative measurable functions