calculus - Equivalent theorems for substitution of variables in limits

I have some doubts on the theorem for substitutions of the variable in limits. There are two equivalent theorems for this.

The first one:

Suppose that $\lim_{x\to c} f(x)=l$ exists whith $c,l\in \mathbb{R} \cup \big\{+\infty,-\infty \big\}$, than let $g$ be a function defined in a neighbourhood $I(l)$ besides (at most) the point $l$, such that:

.If $l\in \mathbb{R}$, $g$ is continous in $l$

.If $l=\big\{+\infty,-\infty \big\}$, the limit $\lim_{y\to l} g(y)$ exists

Then $\lim_{x\to c} g(f(x))=\lim_{y\to l} g(y)$

The second one:

Suppose that $\lim_{x\to c} f(x)=l$ and $\lim_{y\to l} g(y)=m$ exist whith $c,l,m\in \mathbb{R} \cup \big\{+\infty,-\infty \big\}$ and that, if $l\in \mathbb{R}$, there exists a neighbourhood $I(c)$ in which $f(x) \neq l$ $\forall x\in I(c)$ and $x\neq c$

Then again $\lim_{x\to c} g(f(x))=\lim_{y\to l} g(y)$

I don't understand in particular how the second theorem can be equivalent to the first one, and it is not clear to me how the condition for which exists a neighbourhood $I(c)$ in which $f(x) \neq l$ $\forall x\in I(c)$ and $x\neq c$ it is related to existance of the limit.

Can anyone help me with this?

Thanks a lot for your help