Intuitive explanation of sign preservation of limits and boundedness?

Question

Is there an intuitive explanation of the following two statements?

Let $I \subseteq \mathbb{R}$ be an open interval, let $c \in I$ and let $f:I - \left\{c\right\} \to \mathbb{R}$ be a function. If $\lim_{x \to c}f(x)$ exists then there's $\delta >0$ s.t. the restriction of $f$ to $(I - \left\{c\right\}) \cap (c-\delta, c+\delta)$ is bounded.

And with identical assumptions:

Let $I \subseteq \mathbb{R}$ be an open interval, let $c \in I$ and let $f:I - \left\{c\right\} \to \mathbb{R}$ be a function. If $\lim_{x \to c}f(x) >0$ exists then there's $M>0$ and $\delta >0$ s.t. $x \in I - \left\{c\right\} $ and $|x-c| < \delta$ imply $f(x)> M. $

I can follow the proofs. I just don't understand them intuitively.

Answer 1

1. Call the limit $L$. $\lim_{x\to c}f(x)=L$ means intuitively that if $x$ is very close to $c$ (in mathematical language, if $x\in(c-\delta,c+\delta)$ for $\delta$ small), then the values of $f(x)$ are very close to $L$ (in mathematical language, $|f(x)-L|\le\epsilon$.) If $f(x)$ is close to $L$, it cannot be arbitrarily large, meaning that it stays bounded.

2. Now $L>0$. If $f(x)$ is very close to $L$, it cannot be negative.

Hope this helps.