query: Prove that $\lim_{x \to 0} \frac{x - \sin x}{x^3} = \frac16$ using notable limits
positive_passages.docid: 100893
positive_passages.text: Determine $\lim_{x \to 0}{\frac{x-\sin{x}}{x^3}}=\frac{1}{6}$, without L'Hospital or Taylor
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: 78877
negative_passages.text: Limit: $\lim\limits_{x\to0}\frac{e^x-e^{\sin x}}{x-\sin x}$
negative_passages.docid: 113765
negative_passages.text: $\lim_{x\to0} \frac{x-\sin x}{x-\tan x}$ without using L'Hopital
negative_passages.docid: 82968
negative_passages.text: Find $(a,b)$ such that $\lim\limits_{x\to 0}\frac{f(x)}{x}=1$ implies $\lim\limits_{x\to 0}\frac{x(1+a\cos x)-b\sin x}{(f(x))^3}=1$
negative_passages.docid: 143227
negative_passages.text: Show that: $\lim \limits_{n\to\infty}\frac{x_n-x_{n-1}}{n}=0 $
negative_passages.docid: 31146
negative_passages.text: Does the limit $\lim_{(x,y)\to (0,0)} \frac {x^3y^2}{x^4+y^6}$ exist
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: 85249
negative_passages.text: Evaluating: $\lim\limits_{x\to0}\left(\frac{\sin x}{x}\right)^{{6}/{x^{2}}}$
negative_passages.docid: 81537
negative_passages.text: Limit: $\lim_{x\to 0}\frac{\tan3x}{\sin2x}$
negative_passages.docid: 13104
negative_passages.text: What is $\lim_{x \to \infty} \frac{\sin x}{\sin x}?$
negative_passages.docid: 30181
negative_passages.text: Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$
negative_passages.docid: 156610
negative_passages.text: To prove the limit of $\frac{x^2+2\cos x-2}{x\sin^3 x}$ at zero is $1/12$
negative_passages.docid: 12375
negative_passages.text: Prove $\lim_{(x,y) \to (0,0)} \frac{x^3y}{x^4+y^2} = 0$
negative_passages.docid: 123436
negative_passages.text: Precise definition of limits and $\lim_{x\to-1}\frac1{\sqrt{x^2+3}}=\frac12$
negative_passages.docid: 70742
negative_passages.text: Evaluate $\lim_{x\rightarrow 0} \frac{\sin (6x)}{\sin(2x)}$ without L'Hopital
negative_passages.docid: 37755
negative_passages.text: Limit $\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}$