query: Show that the limit $a^x/x \to \infty$ when $a>1$ and $x \to \infty$
positive_passages.docid: 6310
positive_passages.text: How to prove that exponential grows faster than polynomial?
negative_passages.docid: 9781
negative_passages.text: How to prove that $\lim_{n \to \infty} n x^{n} = 0 $ when $0<x<1$?
negative_passages.docid: 122820
negative_passages.text: Prove $\lim_{n\to\infty}x_n=2$ Given $\lim_{n \to \infty} x_n^{x_n} = 4$
negative_passages.docid: 105860
negative_passages.text: Evaluate the limit of $(n+1)\int_0^1x^n\ln(1+x)\,dx$ when $n\to\infty$
negative_passages.docid: 122485
negative_passages.text: Limit of $f(x+\sqrt x)-f(x)$ as $x \to\infty$ if $|f'(x)|\le 1/x$ for $x>1$
negative_passages.docid: 50673
negative_passages.text: Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$
negative_passages.docid: 10428
negative_passages.text: Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist
negative_passages.docid: 44507
negative_passages.text: How can I find $\lim_{n\to \infty} a_n$
negative_passages.docid: 44706
negative_passages.text: Behavior of $\sum a_n x^n$ given $\sum |a_n - a_{n-1}| < \infty$
negative_passages.docid: 73696
negative_passages.text: Given $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $\sum \sqrt{a_n a_{n+1}} < \infty \implies \sum a_n < \infty$
negative_passages.docid: 19512
negative_passages.text: Limit $c^n n!/n^n$ as $n$ goes to infinity
negative_passages.docid: 18156
negative_passages.text: Find $\lim_{x\to \infty} \ln(x+1)/(\ln(x))$
negative_passages.docid: 11539
negative_passages.text: Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$
negative_passages.docid: 42817
negative_passages.text: proving $\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$ when f is continous on [0,1]
negative_passages.docid: 107764
negative_passages.text: Limit series $(a^n-b^n)^{1/n}$
negative_passages.docid: 135700
negative_passages.text: Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists
negative_passages.docid: 65137
negative_passages.text: Evaluating $\lim\limits_{n\to\infty}(a_1^n+\dots+a_k^n)^{1\over n}$ where $a_1 \ge \cdots\ge a_k \ge 0$