query: $\lim_{x\to c}f'(x)=L$ implies $f'(c)=L$
positive_passages.docid: 1371
positive_passages.text: Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
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: 81
negative_passages.text: $\lim_{n\to \infty}f(nx)=0$ implies $\lim_{x\to \infty}f(x)=0$
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: 29287
negative_passages.text: Prove that: $\lim_{n\to\infty} f(n+1) - f(n) = \lim_{x\to\infty} (f(x))' $
negative_passages.docid: 5229
negative_passages.text: Prove that $\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$ is equal to $f(1)$.
negative_passages.docid: 10431
negative_passages.text: Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$
negative_passages.docid: 147203
negative_passages.text: How to show that $\lim_{n\to\infty}\left\{f(c+\frac{1}{n^2})+f(c+\frac{2}{n^2})+\cdots+f(c+\frac{n}{n^2})-nf(c)\right\}=\frac{1}{2} f'(c)$
negative_passages.docid: 6620
negative_passages.text: If $\lim\limits_{x\rightarrow\infty} (f'(x)+f(x)) =L<\infty$, does $\lim\limits_{x\rightarrow\infty} f(x) $ exist?
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
negative_passages.docid: 44336
negative_passages.text: Is $\lim\limits_{x\to x_0}f'(x)=f'(x_0)$?
negative_passages.docid: 19986
negative_passages.text: How can this be proved $\lim_{x\to\infty}(f(x)+f'(x))=l$
negative_passages.docid: 11879
negative_passages.text: If $\lim_{n\rightarrow\infty} f(n+1) - f(n) = L$, prove that $\lim_{n\rightarrow\infty} f(n)/n = L$
negative_passages.docid: 143354
negative_passages.text: Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
negative_passages.docid: 26424
negative_passages.text: If $f(x)<g(x)$ prove that $\lim f(x)<\lim g(x)$
negative_passages.docid: 72960
negative_passages.text: How to prove this limit composition theorem?
negative_passages.docid: 59724
negative_passages.text: Proving that $x_n\to L$ implies $|x_n|\to |L|$, and what about the converse?