query: Prove $\lim_{x\to 0} \frac{ 1 -\cos x } {x^2} =\frac1 2$
positive_passages.docid: 36063
positive_passages.text: Finding the limit of $(1-\cos x)/x^2$
negative_passages.docid: 147184
negative_passages.text: $f:[-1,1]\to \mathbb{R}$ is a continuous function such that $f(2x^2-1)=(x^3+x)f(x).$ Find $\lim_{x\to 0}\frac {f(\cos x)}{\sin x}$
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: 60321
negative_passages.text: Use the definition of derivative to prove $\lim_{x\to 0}\ln(x+1)/x =1$
negative_passages.docid: 155979
negative_passages.text: $\lim_{x\to 0} \frac{\sin x - x}{x^2}$ without L'Hospital or Taylor
negative_passages.docid: 94168
negative_passages.text: evaluate the limit as $x$ approaches $0$ of $(\cos x-1)/x$ without l'Hopital's rule
negative_passages.docid: 97326
negative_passages.text: Find $\lim_{x \to 0} (\frac{\tan(x)}{x})^{\frac{1}{x}}$
negative_passages.docid: 145496
negative_passages.text: Show $\lim_{h \to \ 0} \frac{f(x + 2h) - 2f(x+h) + f(x)}{h^{2}} = f''(x)$ Proof verification
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: 65239
negative_passages.text: Find $\lim_{x \to 0} \frac{x - \tan(x)}{x - \sin(x)}$
negative_passages.docid: 133190
negative_passages.text: Calculus - Prove $\lim_{n \to \infty} \frac{x^n}{n!}=0$
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: 106281
negative_passages.text: Prove $\lim\limits_{n\to \infty}\frac{1}{\sqrt n}\left|\sum\limits_{k=1}^n (-1)^k\sqrt k\right|= \frac{1}{2}$
negative_passages.docid: 3110
negative_passages.text: Prove $\frac{1}{\sqrt{x}}\geq \frac{\ln x}{x-1}$
negative_passages.docid: 128063
negative_passages.text: How to evaluate the limit $\lim_{x\to 0} \frac{1-\cos(4x)}{\sin^2(7x)}$
negative_passages.docid: 154264
negative_passages.text: $f \in C[-1,1]$, Prove ${\lim_{h \to 0^+}}{\int_{-1}^1 \frac{h}{h^2+x^2}f(x)\,dx} = \pi f(0)$
negative_passages.docid: 126256
negative_passages.text: Prove limit of function