query: Solving $\lim_{t\rightarrow 0}\frac{1}{\pi}\int_{-1}^1 f(x) \frac{t}{t^2+x^2} dx =f(0)$ without dominated convergence
positive_passages.docid: 154264
positive_passages.text: $f \in C[-1,1]$, Prove ${\lim_{h \to 0^+}}{\int_{-1}^1 \frac{h}{h^2+x^2}f(x)\,dx} = \pi f(0)$
negative_passages.docid: 156478
negative_passages.text: Prove that $\lim_{n\rightarrow\infty}\frac{x_1^2+x_2^2+\cdots+x_n^2}{n^2}=0$
negative_passages.docid: 14385
negative_passages.text: Solve $\int \limits_{0}^{\infty} \frac{\cos(x)}{\cosh(x)} dx$ without complex integration.
negative_passages.docid: 55173
negative_passages.text: Solve for $\lim_{x \rightarrow \infty} \frac{x^{\ln(x)}}{(\ln(x))^x}$
negative_passages.docid: 36487
negative_passages.text: Proving $\frac{1}{n^2}$ infinite series converges without integral test
negative_passages.docid: 98365
negative_passages.text: How to evaluate $\lim_{x \rightarrow \infty} \left(\left(\frac{x+1}{x-1}\right)^x -e^2\right){x}^2$
negative_passages.docid: 20956
negative_passages.text: Correct way to prove the limit: $\mathop {\lim }\limits_{x \to 0} {x^\alpha }\int\limits_x^1 {\frac{{f\left( t \right)}}{{{t^{\alpha  + 1}}}}dt} $?
negative_passages.docid: 60831
negative_passages.text: Need hint for $\lim_{x\to 0} \frac{(x+1)^\frac{1}{x}-e}{x}$
negative_passages.docid: 10093
negative_passages.text: $\lim_{x\to \infty}e^{-x^2}\int_x^{x+1/x}e^{t^2}dt$
negative_passages.docid: 92176
negative_passages.text: Solving $\lim_{n\to\infty}(n\int_0^{\pi/4}(\tan x)^ndx)$?
negative_passages.docid: 52943
negative_passages.text: How to show $\lim_{\epsilon \rightarrow 0} \frac{\epsilon}{x^2+\epsilon ^2} = \pi \delta (x)$?
negative_passages.docid: 152640
negative_passages.text: show that $\lim(\pi(x)/x) = 0$
negative_passages.docid: 79618
negative_passages.text: Find the limit without use of L'Hôpital or Taylor series: $\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right)$
negative_passages.docid: 135565
negative_passages.text: Limit of $\frac{1}{x^2}-\frac{1}{\sin^2(x)}$ as $x$ approaches $0$
negative_passages.docid: 104269
negative_passages.text: Why does $\lim\limits_{t\to-\infty}te^t=0$?
negative_passages.docid: 3200
negative_passages.text: Limit of integral without Taylor expansion
negative_passages.docid: 49860
negative_passages.text: Determine the convergence of $\sum\limits_{n=2}^\infty \frac{\sqrt{n+1}}{n(n-1)}$