query: If $f \in C^\infty$, then $f''(x) = \lim_{t \to 0} \frac{f(x + t) + f(x - t) - 2f(x)}{t^2}$
positive_passages.docid: 145496
positive_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: 132824
negative_passages.text: Does $f$ monotone and $f\in L_{1}([a,\infty))$ imply $\lim_{t\to\infty} t f(t)=0$?
negative_passages.docid: 20404
negative_passages.text: Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$
negative_passages.docid: 133190
negative_passages.text: Calculus - Prove $\lim_{n \to \infty} \frac{x^n}{n!}=0$
negative_passages.docid: 20675
negative_passages.text: $L=\lim_{x\to\infty}(f(x)+f'(x))$ exists . Which of the following statements is\are correct?
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: 54830
negative_passages.text: Find the limit $\lim\limits_{x\to\infty} f(x)= \lim\limits_{x \to \infty} \left(\frac{x}{x+1}\right)^x$
negative_passages.docid: 40564
negative_passages.text: Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$
negative_passages.docid: 86409
negative_passages.text: Show that $\lim_{x \to +\infty}\left(f(x)+f'(x)\right)=0 \Rightarrow \lim_{x \to +\infty} f(x)=0$
negative_passages.docid: 109489
negative_passages.text: How to solve $ \lim_{n\to\infty} \frac {(n!)^\frac{1}{n}}{n} $?
negative_passages.docid: 44370
negative_passages.text: If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$
negative_passages.docid: 50472
negative_passages.text: If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$
negative_passages.docid: 130161
negative_passages.text: Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$
negative_passages.docid: 74170
negative_passages.text: Prove that $\lim_{x \to +\infty} \frac{f(x)}{x} = L$ if $\lim_{x \to +\infty} [f(x+1) - f(x)] = L \space$
negative_passages.docid: 138544
negative_passages.text: For which values of $x$, will $f (x) . f '(x) < 0$?
negative_passages.docid: 36358
negative_passages.text: How prove that $\lim\limits_{x\to+\infty}f(x)=\lim\limits_{x\to+\infty}f'(x)=0$ if $\lim\limits_{x\to+\infty}([f'(x)]^2+f^3(x))=0$?
negative_passages.docid: 127125
negative_passages.text: Can the limit $\lim_{h \to 0}\frac{f(x + h) - 2f(x) + f(x - h)}{h^2}$ exist if $f'(x)$ does not exist at $x$?