query: Prove that $\lim_{x \to 0} \frac{f(x)}{x}=0$
positive_passages.docid: 25467
positive_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: 41037
negative_passages.text: Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions
negative_passages.docid: 129042
negative_passages.text: Prove that $\lim_{a \to 0^{+}} \int_{0}^{a} \frac{1}{\sqrt{\cos(x)-\cos(a)}} \;dx=\frac{\pi}{\sqrt{2}}$
negative_passages.docid: 47606
negative_passages.text: Proving $\lim_{n\to\infty} a^{\frac{1}{n}}=1$ by definition of limit
negative_passages.docid: 149586
negative_passages.text: If $\lim_{x\to\infty} f'(x) = L$, prove that $\lim_{x\to\infty} f(x+c)-f(x) = cL$ and $\lim_{x\to\infty} f(x)/x=L$
negative_passages.docid: 60878
negative_passages.text: How to prove that $\lim_{(x,y) \to (0,0)} \frac{x^3y}{x^4+y^2} = 0?$
negative_passages.docid: 105527
negative_passages.text: Show that $\lim_{r\to 0} \frac{1}{r^2}\int_{C_{r}}f(z)dz = 2 \pi i\frac{\partial f}{\partial \bar{z}}(z_0)$
negative_passages.docid: 133190
negative_passages.text: Calculus - Prove $\lim_{n \to \infty} \frac{x^n}{n!}=0$
negative_passages.docid: 71225
negative_passages.text: $f(0)=0$ and $\lim_{x \to \infty}f(x)=0$. Show there's $x_0$ such that $f'(x_0)=0$
negative_passages.docid: 1057
negative_passages.text: How prove this integral limit $=f(\frac{1}{2})$
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: 22709
negative_passages.text: Show that $\lim_{x\to \frac{\pi}{2}} \frac{1}{\big(x-\frac{\pi}{2}\big)}+{\tan(x)}=0$.
negative_passages.docid: 51212
negative_passages.text: How do I solve $\lim\limits_{x\to\infty} \ \frac{\ln \ x}{x} = 0$?
negative_passages.docid: 19263
negative_passages.text: Limit as $x\to 0$ of $\frac{(1+x)^{1/x}-e}{x}$
negative_passages.docid: 60831
negative_passages.text: Need hint for $\lim_{x\to 0} \frac{(x+1)^\frac{1}{x}-e}{x}$
negative_passages.docid: 98392
negative_passages.text: Value of the limit $\lim_{x \to 0} \frac{\sin(\frac{1}{x})}{\sin(\frac{1}{x})}$
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