query: I want to know rigorous proof of lim x→0 sin(x)/x=1
positive_passages.docid: 17029
positive_passages.text: How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
negative_passages.docid: 127767
negative_passages.text: How to find $\lim_{(x,y)\rightarrow (0,0)} \frac{\sin(x\cdot y)}{x}$?
negative_passages.docid: 126701
negative_passages.text: How to prove that $d \sin(x)/dx = \cos(x)$ without circular logic such as L'Hôpital's rule?
negative_passages.docid: 125087
negative_passages.text: Evaluate the limit of ratio of sums of sines (without L'Hopital): $\lim_{x\to0} \frac{\sin x+\sin3x+\sin5x}{\sin2x+\sin4x+\sin6x}$
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: 37755
negative_passages.text: Limit $\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}$
negative_passages.docid: 28161
negative_passages.text: Solve $\lim_{x\to 0} \frac{\sin x-x}{x^3}$
negative_passages.docid: 72960
negative_passages.text: How to prove this limit composition theorem?
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: 93155
negative_passages.text: Evaluate $\mathop {\lim }\limits_{x \to 0} \left( {{1 \over {{{\sin }^2}x}} - {1 \over {{x^2}}}} \right)$
negative_passages.docid: 30181
negative_passages.text: Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$
negative_passages.docid: 25467
negative_passages.text: Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$
negative_passages.docid: 79684
negative_passages.text: Finding $\lim \limits_{x \to 0} \frac{1 - \cos x}{x}$, given $\lim \limits_{x \to 0} \frac{\sin x}{x} = 1$
negative_passages.docid: 78877
negative_passages.text: Limit: $\lim\limits_{x\to0}\frac{e^x-e^{\sin x}}{x-\sin x}$
negative_passages.docid: 65975
negative_passages.text: Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)$
negative_passages.docid: 68700
negative_passages.text: Proving Two Limits Are Equal to Each Other
negative_passages.docid: 5947
negative_passages.text: Limit of a trigonometric rational expression