In Hubbard's Vector Calculus, I saw an unconventional definition for the limit of a function:
Definition: Let $X$ be a subset of $\mathbb{R}^n$. A function $\mathbf{f}:X\rightarrow\mathbb{R}^m$ has the limit $\mathbf{a}$ at $\mathbf{x_0}$:
$$ \lim_{\mathbf{x} \to \mathbf{x_0}} \mathbf{f(x)} = \mathbf{a} $$ if $\mathbf{x_0}$ is in the closure of $X$, and for every $\epsilon>0$, there exists $\delta>0$ such that for all $\mathbf{x}\in X$, $|\mathbf{x} - \mathbf{x_0}| < \delta$ implies $|\mathbf{f(x)} - \mathbf{a}| < \epsilon$.
The peculiarity of the definition is the usage of "$|\mathbf{x} - \mathbf{x_0}| < \delta$" instead of $0<|\mathbf{x} - \mathbf{x_0}| < \delta$.
I think this subtle difference can have rather significant implications. Using this definition, if $ \lim_{\mathbf{x} \to \mathbf{x_0}} \mathbf{f(x)}$ exists, then $\mathbf{f(x)}$ is continuous at $\mathbf{x_0}$, right? Are there other profound implications you can think of?
No, there aren't really, since you hit the nail on the head. This alternate definition of a limit is the standard definition of continuity. So any other implications of this alternate definition will stem from the fact that limits have been redefined as continuity.
You could say that one implication is that the intuitive meaning of a limit no longer holds. For example, define $f: \mathbb{R} \to \mathbb{R}$ by $f(x) = 0$ for $x = 0$ and $f(x) = 1$ for all $x \ne 0$. The alternate definition would say that there is no limit at $x = 0$. (Normally, you would say that $f$ is not continuous at $x = 0$.