query: $\lim_{z \to a} f(z) = L$ iff $\forall z_n s.t. z_n \to a$ as $n \to \infty, f(z_n) \to L$
positive_passages.docid: 22989
positive_passages.text: A doubt on the definition of limit of a function definition
negative_passages.docid: 39034
negative_passages.text: How would you evaluate $\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$
negative_passages.docid: 99906
negative_passages.text: If $f'(c)$ exists then $\lim_{x\to c} f(x)=f(c)$
negative_passages.docid: 135877
negative_passages.text: $\lim\limits_{n \to \infty} \frac{a_n}{b_n}=l$ ; Prove $\sum\limits_{k=1}^\infty a_k$ converges iff $\sum\limits_{k=1}^\infty b_k$ converges.
negative_passages.docid: 152132
negative_passages.text: Proving if $\lim_{n\rightarrow\infty}a_n=L $ then $\lim_{n\rightarrow\infty} \frac{a_1+a_2+\cdots+a_n}n=L $
negative_passages.docid: 33304
negative_passages.text: Let $f$ be a continuous function satisfying $\lim \limits_{n \to \infty}f(x+n) = \infty$ for all $x$. Does $f$ satisfy $f(x) \to \infty$?
negative_passages.docid: 144165
negative_passages.text: $n\in N, a_{n+1} = a_n (1 - a_n)$ , $0 < a_0 < 1$. Prove that $\lim\limits_{n\to\infty} a_n\cdot n = 1$.
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: 48593
negative_passages.text: Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$
negative_passages.docid: 113141
negative_passages.text: Prove that $\lim_{n \to \infty}\bigg[\int_0^1f(t)^n \text{dt}\bigg]^{1/n}=M$
negative_passages.docid: 67084
negative_passages.text: Prove or disprove that $\lim_{n \to \infty} {n \choose k} / a^n = 0$ for all $k \in \mathbb{Z_+}, a \in \mathbb{R}: a> 1$.
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: 155333
negative_passages.text: Prove that $\lim_{x\to\infty}\frac{f(x)}x=\lim_{x\to\infty}f'(x)$
negative_passages.docid: 73812
negative_passages.text: Does $\lim \limits_ {n\to \infty}\ \frac{n}{n+1}$ equal $1$?
negative_passages.docid: 49497
negative_passages.text: $\lim a_n = 0$ if $\left|\frac{a_{n +1}}{a_n}\right|\to L < 1$
negative_passages.docid: 119200
negative_passages.text: When does $\lim_{n\to\infty}f(x+\frac{1}{n})=f(x)$ a.e. fail
negative_passages.docid: 67364
negative_passages.text: If $\lim_{x\to\infty}(f(x)+f'(x))=L$ show that $\lim_{x\to\infty} f(x) = L$ and $\lim_{x\to\infty} f'(x) = 0$