query: Why $\lim_{x \to \infty} (1 + \frac{1}{x})^x = e$?
positive_passages.docid: 64169
positive_passages.text: Why isn't $\lim \limits_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}$ equal to $1$?
negative_passages.docid: 77752
negative_passages.text: Show that $e^{x+y}=e^xe^y$ using $e^x=\lim_{n\to\infty }\left(1+\frac{x}{n}\right)^n$.
negative_passages.docid: 26338
negative_passages.text: Does $1^{\infty}=e$ or $1^{\infty}=1$?
negative_passages.docid: 21380
negative_passages.text: Proof for $e^z = \lim \limits_{x \rightarrow \infty} \left( 1 + \frac{z}{x} \right)^x$
negative_passages.docid: 67894
negative_passages.text: $\epsilon$-$\delta$ proof that $\lim\limits_{x \to 1} \frac{1}{x} = 1$.
negative_passages.docid: 58456
negative_passages.text: Why is $ 1^{\infty} \not= \lim_{t \to \infty} 1^t = 1$?
negative_passages.docid: 135731
negative_passages.text: Is there a way to show that $e^x=\lim_{n\to \infty }\left(1+\frac{x}{n}\right)^n$?
negative_passages.docid: 15292
negative_passages.text: Elementary proof for $\lim\limits_{n \to\infty}\frac{n!e^n}{n^n} = +\infty$
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: 114747
negative_passages.text: If $\lim_{x \to \infty}f'(x)=+\infty$ then $\lim_{x \to \infty}(f(x)-f(x-1))=+\infty$ and $\lim_{x \to \infty}f(x)=+\infty$.
negative_passages.docid: 30160
negative_passages.text: How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$?
negative_passages.docid: 101716
negative_passages.text: Find this limit: $ \lim_{n \to \infty}{(e^{\frac{1}{n}} - \frac{2}{n})^n}$
negative_passages.docid: 71798
negative_passages.text: $\lim_{n\to \infty}\frac{1}{n} \sum_{k=1}^n x_k =x\;$ given $\;\lim_{n\to \infty} x_n= x\;?$
negative_passages.docid: 91680
negative_passages.text: Does $\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x)\,dx\,$?
negative_passages.docid: 97747
negative_passages.text: Limit of $\lim\limits_{n\to\infty} (1 + \frac{x_n}{n})^n$
negative_passages.docid: 144273
negative_passages.text: Find $\lim_{n\to\infty} \frac{1}{n} \int_0^\infty f(x) e^{-a_n x} dx$
negative_passages.docid: 138169
negative_passages.text: Finding $\lim\limits_{n \to \infty }n\int_{0}^{1}\frac{x^{n}}{1+x+x^{n}}dx$