query: calculate $ \lim \limits_{x \to 0} x^a \int_x^1 \frac{f(t)}{t^{a+1}} \, dt$
positive_passages.docid: 20956
positive_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: 135159
negative_passages.text: Limits using L'Hopital's rule $\lim_{x\to0^+} (x^{x^x-1})$
negative_passages.docid: 34790
negative_passages.text: Find $f(1)$ from equation $\int_{0}^{x}f(t)dt = x+\int_{x}^{1}tf(t)dt$
negative_passages.docid: 59890
negative_passages.text: Fundamental Theorem of Calculus for $\lim\limits_{x\to 0}\frac{\int_0^x(x-t)\sin t^2\ dt}{x\sin^3x}$
negative_passages.docid: 113289
negative_passages.text: finding the limit $\lim\limits_{x \to \infty }(\frac{1}{e}(1+\frac{1}{x})^x)^x$
negative_passages.docid: 86843
negative_passages.text: Find the value of : $\lim_{n\to\infty}[(n+1)\int_{0}^{1}x^{n}\ln(1+x)dx]$.
negative_passages.docid: 19263
negative_passages.text: Limit as $x\to 0$ of $\frac{(1+x)^{1/x}-e}{x}$
negative_passages.docid: 19631
negative_passages.text: Finding $\lim_{n\to \infty}\sqrt n \int_0^1 \frac{\,dx}{(1+x^2)^n}$
negative_passages.docid: 156231
negative_passages.text: Computing $\lim\limits_{n\to\infty} \int_{0}^{\infty}\frac{\sin(x/n)}{(1+x/n)^{n}}\, dx$
negative_passages.docid: 56671
negative_passages.text: Find the limit $ \lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}} $
negative_passages.docid: 10456
negative_passages.text: Finding limits $l=\lim_{x \rightarrow a}\frac{x^x-a^x}{x-a} $ & $m= \lim_{x \rightarrow a}\frac{a^x-x^a}{x-a} $.
negative_passages.docid: 95508
negative_passages.text: Evaluate $\lim\limits_{x\to 1}\frac{\sqrt x-x^2 }{1-\sqrt x}$
negative_passages.docid: 128456
negative_passages.text: Evaluating $\lim_{x\to0}\left(\frac{1+x2^x}{1+x3^x}\right)^\frac1{x^2}$
negative_passages.docid: 17090
negative_passages.text: Compute $\lim_\limits{n\to\infty}a_n$ where $a_{n+2}=\sqrt{a_n.a_{n+1}}$
negative_passages.docid: 37394
negative_passages.text: How to find $\lim\limits_{n\to\infty} n·(\sqrt[n]{a}-1)$?
negative_passages.docid: 56085
negative_passages.text: Calculate the limit : $\lim_{x \to 0}\frac{x-\sin{x}}{x^3}$ WITHOUT using L'Hopital's rule
negative_passages.docid: 138169
negative_passages.text: Finding $\lim\limits_{n \to \infty }n\int_{0}^{1}\frac{x^{n}}{1+x+x^{n}}dx$