query: let $f: J \to \mathbb{R}$ . let $x_n < c <y_n$ be such that $(y_n-x_n) \to0$ show that $\lim _ {n \to \infty} \frac{f(y_n)-f(x_n)}{y_n-x_n}=f'(c)$
positive_passages.docid: 20280
positive_passages.text: Proving that expression is equivalent to the definition of derivative
negative_passages.docid: 48891
negative_passages.text: Show that: lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$
negative_passages.docid: 84818
negative_passages.text: Convergence of $\sum_{n=1}^{\infty}x_n\overline{y_n}$ for every $(x_n)\in c_0$ implies $\sum_{n=1}^{\infty}|y_n|<\infty$
negative_passages.docid: 88147
negative_passages.text: Show that $g_n(x)=nf(x+1/n)-nf(x)$ converges uniformly to $f'$
negative_passages.docid: 70746
negative_passages.text: Proving that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence.
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: 7507
negative_passages.text: Proof that $\lim{a_n}=L$ when $n$ goes to infinity, then $\{a_n\}$ its a Cauchy Sequence
negative_passages.docid: 97747
negative_passages.text: Limit of $\lim\limits_{n\to\infty} (1 + \frac{x_n}{n})^n$
negative_passages.docid: 37489
negative_passages.text: Show that $ \lim_{(x,y) \to 0} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \alpha/\gamma + \beta/\delta > 1.$
negative_passages.docid: 2449
negative_passages.text: Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
negative_passages.docid: 110126
negative_passages.text: Prove that $\lim\limits_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$
negative_passages.docid: 72960
negative_passages.text: How to prove this limit composition theorem?
negative_passages.docid: 119804
negative_passages.text: If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .
negative_passages.docid: 104456
negative_passages.text: Show that $\lim_{n\to\infty}b_n=\frac{\sqrt{b^2-a^2}}{\arccos\frac{a}{b}}$
negative_passages.docid: 4289
negative_passages.text: $\lim_{x\to0}\dfrac{1}{x}-\lim_{x\to0}\dfrac{1}{x}$
negative_passages.docid: 145903
negative_passages.text: Prove that $\int_E |f_n-f|\to0 \iff \lim\limits_{n\to\infty}\int_E|f_n|=\int_E|f|.$
negative_passages.docid: 19986
negative_passages.text: How can this be proved $\lim_{x\to\infty}(f(x)+f'(x))=l$