query: If $\lim_{x\to c}f'(x)=A$ then $f'(c)=A$
positive_passages.docid: 1371
positive_passages.text: Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
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: 128218
negative_passages.text: If $\lim \limits_{x \to a}\left(f(x)+\frac{1}{f(x)}\right)=2,$ then $\lim \limits_{x \to a}f(x)=1$
negative_passages.docid: 138930
negative_passages.text: What does $\lim\limits_{x \to \infty} f(x) = 1$ say about $\lim\limits_{x \to \infty} f'(x)$?
negative_passages.docid: 127921
negative_passages.text: Does there exist a function such that $\lim_{x \to a} f(x) = L$ for all $a \in \mathbb R$ but $f(x)$ is never $L$?
negative_passages.docid: 53236
negative_passages.text: Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$
negative_passages.docid: 42817
negative_passages.text: proving $\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$ when f is continous on [0,1]
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: 155333
negative_passages.text: Prove that $\lim_{x\to\infty}\frac{f(x)}x=\lim_{x\to\infty}f'(x)$
negative_passages.docid: 13486
negative_passages.text: Show $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \|f\|_{\infty}$ for $f \in L^{\infty}$
negative_passages.docid: 23222
negative_passages.text: If $f(x) + f'(x) + f''(x) \to A$ as $x \to \infty$, then show that $f(x) \to A$ as $x \to \infty$
negative_passages.docid: 29287
negative_passages.text: Prove that: $\lim_{n\to\infty} f(n+1) - f(n) = \lim_{x\to\infty} (f(x))' $
negative_passages.docid: 117906
negative_passages.text: Show that $\liminf\limits_{x\to \infty}|f'(x)|=0$ if $\lim\limits_{x\to \infty}f(x)/x=0$ and $f$ is differentiable
negative_passages.docid: 135577
negative_passages.text: Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
negative_passages.docid: 74673
negative_passages.text: True or false: if $f'(c)<0$, then $f$ is concave down at $x=c$?
negative_passages.docid: 78834
negative_passages.text: if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
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