query: $f(x)$ is differentiable on $\mathbb{R}$, If $\lim_{x \to \infty}f'(x)=a$,then show $\lim_{x \to \infty}(f(x+1) - f(x))=a$
positive_passages.docid: 149586
positive_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: 99906
negative_passages.text: If $f'(c)$ exists then $\lim_{x\to c} f(x)=f(c)$
negative_passages.docid: 80643
negative_passages.text: Show that f(x) defined on [0,infinity) R to is continuous at 0.
negative_passages.docid: 136412
negative_passages.text: Differentiable at $x=a$ implies continuous at $x=a$
negative_passages.docid: 143874
negative_passages.text: Limit of derivative is zero. Does it imply a limit for f(x)?
negative_passages.docid: 49929
negative_passages.text: If $ \lim_{n \to \infty} (2 x_{n + 1} - x_{n}) = x $, then is it true that $ \lim_{n \to \infty} x_{n} = x $?
negative_passages.docid: 59837
negative_passages.text: Suppose $\mu(X)<\infty$. Show that if $\ f\in\ L^q(X,\mu)$ for some $1\le q<\infty$ then $\ f\in\ L^r(X,\mu)$ for $1\le r<q$
negative_passages.docid: 105068
negative_passages.text: Proving that the function is differentiable and $f'(0)=0$
negative_passages.docid: 121217
negative_passages.text: How to show that $\lim\limits_{x \to \infty} f'(x) = 0$ implies $\lim\limits_{x \to \infty} \frac{f(x)}{x} = 0$?
negative_passages.docid: 57695
negative_passages.text: Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
negative_passages.docid: 38923
negative_passages.text: What can we say about differentiability of $f$ at $c$?
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: 137527
negative_passages.text: if $f(x)$ is differentiable at a x, prove that: $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$
negative_passages.docid: 31685
negative_passages.text: Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$
negative_passages.docid: 19986
negative_passages.text: How can this be proved $\lim_{x\to\infty}(f(x)+f'(x))=l$
negative_passages.docid: 119602
negative_passages.text: $f:\mathbb{R} \to \mathbb{R}$ be differentiable and $\lim\limits_{x\to\infty}f'(x)=1$, is $f(x)$ unbounded?
negative_passages.docid: 142490
negative_passages.text: Alternative proof for differentiability of inverse function?